Concept Builders - A Complete Guide To Entrance Test Examination: Math

Shortcut Method To Find Derivative of Implicit Functions

<div dir="ltr" style="text-align: left;" trbidi="on"><div dir="ltr" style="text-align: left;" trbidi="on"><div class="separator" style="clear: both; text-align: center;"></div><div style="text-align: left;">Assalam O Alaikum !</div><div style="text-align: left;">                                 Hi Guys, How are you ? I hope you are enjoying a good health. I am back with a new lesson for you. Well, I know it is quite easy to find the derivative of Implicit functions but it takes too much time. I am gonna tell you a shortcut method to find derivative of implicit functions which will surely save your time. </div> 
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Shortcut Method To Find Derivative of Implicit Functions

Assalam O Alaikum ! Hi Guys, How are you ? I hope you are enjoying a good health. I am back with a new less...

How To Find The Rank of Matrix

<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: left;">The Rank of a Matrix :</div><div style="text-align: left;">You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as a set of c column vectors, each having r elements.</div><div style="text-align: left;"> 
The rank of a matrix is defined as : 
(a) the maximum number of linearly independent column vectors in the matrix 
or 
(b) the maximum number of linearly independent row vectors in the matrix. 
Both definitions are equivalent.</div><div style="text-align: left;"> 
For an r x c matrix,</div><ul style="text-align: left;"><li>If r is less than c, then the maximum rank of the matrix is r.</li>
<li>If r is greater than c, then the maximum rank of the matrix is c.</li>
</ul>The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one. 
 
 
<div style="text-align: left;">How to Find Matrix Rank :</div><div style="text-align: left;"> 
In this section, we describe a method for finding the rank of any matrix. This method assumes familiarity with echelon matrices and echelon transformations. 
The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows. 
Consider matrix A and its row echelon matrix, Aref. <table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; margin-top: 20px; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">7</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">8</td></tr>
</tbody></table></td><td style="margin: 0px; padding: 0px;">   ⇒   </td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td></tr>
</tbody></table></td></tr>
<tr><td style="margin: 0px; padding: 0px; text-align: center; vertical-align: bottom;">A</td><td style="margin: 0px; padding: 0px;"></td><td style="margin: 0px; padding: 0px; text-align: center; vertical-align: bottom;">Aref</td></tr>
</tbody></table>Because the row echelon form Aref has two non-zero rows, we know that matrix A has two independent row vectors; and we know that the rank of matrix A is 2. 
You can verify that this is correct. Row 1 and Row 2 of matrix A are linearly independent. However, Row 3 is a linear combination of Rows 1 and 2. Specifically, Row 3 = 3*( Row 1 ) + 2*( Row 2). Therefore, matrix A has only two independent row vectors.</div><div style="text-align: left;"> 
Full Rank Matrices :</div><div style="text-align: left;"> 
When all of the vectors in a matrix are linearly independent, the matrix is said to be full rank. Consider the matrices A and B below.<table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px 0px 10px;">A =  </td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-bottom: 10px; margin-left: 10px; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdrightbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">6</td></tr>
</tbody></table></td><td style="margin: 0px; padding: 0px; width: 60px;"> </td><td style="margin: 0px; padding: 0px 0px 10px;">B =  </td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-bottom: 10px; margin-left: 10px; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td></tr>
</tbody></table></td></tr>
</tbody></table>Notice that row 2 of matrix A is a scalar multiple of row 1; that is, row 2 is equal to twice row 1. Therefore, rows 1 and 2 are linearly dependent. Matrix A has only one linearly independent row, so its rank is 1. Hence, matrix A is not full rank. 
Now, look at matrix B. All of its rows are linearly independent, so the rank of matrix B is 3. Matrix B is full rank.</div><div style="text-align: left;"> 
Test Your Understanding:</div><div style="text-align: left;"> 
Problem 1 :Consider the matrix X, shown below.<table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;">X =    </td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdrightbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">8</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">0</td></tr>
</tbody></table></td></tr>
</tbody></table>What is its rank? 
(A) 0 
(B) 1 
(C) 2 
(D) 3 
(E) 4 
 
</div><div style="text-align: left;">Solution :The correct answer is (C). Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer rows than columns, its maximum rank is equal to the maximum number of linearly independent rows. And because neither row is linearly dependent on the other row, the matrix has 2 linearly independent rows; so its rank is 2. 
 
</div><div style="text-align: left;">Problem 2Consider the matrix Y, shown below.<table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;">Y =    </td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdrightbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">5</td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">7</td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">5</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">9</td></tr>
</tbody></table></td></tr>
</tbody></table>What is its rank? 
(A) 0 
(B) 1 
(C) 2 
(D) 3 
(E) 4 
 
</div><div style="text-align: left;">SolutionThe correct answer is (C). Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer columns than rows, its maximum rank is equal to the maximum number of linearly independent columns. 
Columns 1 and 2 are independent, because neither can be derived as a scalar multiple of the other. However, column 3 is linearly dependent on columns 1 and 2, because column 3 is equal to column 1 plus column 2. That leaves the matrix with a maximum of two linearly independent columns; that is, column 1 and column 2. So the matrix rank is 2.</div><div style="font-weight: normal; text-align: left;"> 
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How To Find The Rank of Matrix

The Rank of a Matrix : You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as...

Shortcut To Find Rank of Matrix

<div dir="ltr" style="text-align: left;" trbidi="on"><div style="background-color: white; font-stretch: normal; line-height: 20px; margin: 20px 0px 0px; padding: 0px; vertical-align: top;"><div style="font-stretch: normal; margin: 20px 0px 0px; padding: 0px; vertical-align: top;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJIl-SujBxI9LFtvseeLru6WBh5NagHVoT6Pw9_OVs5pLUjP0TXGtbMbwRoI6gkoGyC-j9ypXLbagCgEjlJptR2g_6Cc1NnlZdyQTuiLeWdaJnrzKeXmtkpEispjcvSBKsIShX0KEB15Y/s1600/wanga14.png" imageanchor="1" style="font-size: 24px; line-height: normal; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" height="216" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJIl-SujBxI9LFtvseeLru6WBh5NagHVoT6Pw9_OVs5pLUjP0TXGtbMbwRoI6gkoGyC-j9ypXLbagCgEjlJptR2g_6Cc1NnlZdyQTuiLeWdaJnrzKeXmtkpEispjcvSBKsIShX0KEB15Y/s320/wanga14.png" width="320" /></a></div><div style="font-stretch: normal; margin: 20px 0px 0px; padding: 0px; vertical-align: top;"></div><div style="font-stretch: normal; margin: 20px 0px 0px; padding: 0px; vertical-align: top;">Shortcut To Find Rank Of Matrix :</div><ul style="line-height: 19.2000007629395px;"><li>Look at the matrix whether it is rectangular or square matrix.</li>
</ul><ul style="line-height: 19.2000007629395px;"><li>For rectangular matrix, if number of rows is less than number of columns then the rank of matrix will be equal to number of linearly independent rows.Similarly, If number of columns is less than number of rows then rank of matrix will be equal to number of linearly independent columns.</li>
</ul><ul style="line-height: 19.2000007629395px;"><li>For square matrix, number of linearly independent rows or columns is called rank of matrix.</li>
</ul><table style="border-collapse: collapse; border-spacing: 0px; color: black; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;">Y =    </td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">1</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdrightbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">2</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">5</td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">3</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">7</td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">4</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">5</td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;">9</td></tr>
</tbody></table></td></tr>
</tbody></table>Now, You can see, column 1 and 2 are independent because they are not derived form others. but column 3 (C1 + C2) is sum of column 1 and column 2. So there are two linearly independent columns hence its rank is 2. 
<h2 style="font-stretch: normal; line-height: normal; margin: 20px 0px 0px; padding: 0px; vertical-align: top;"><div style="font-weight: normal; line-height: 19.2000007629395px;">Now, most of us don't know what is meant by linearly independent rows or columns.</div><div style="font-weight: normal; line-height: 19.2000007629395px;"> 
</div><div style="line-height: 19.2000007629395px;">Linearly Independent Rows/Columns : The rows/columns which is not derived from other rows/columns (scalar multiple of other rows/columns or sum of two rows/columns) i.e. which don't depend on other rows/columns.</div><div style="font-weight: normal; line-height: 19.2000007629395px;">For example : </div></h2><h2 style="font-stretch: normal; font-weight: normal; line-height: normal; margin: 20px 0px 0px; padding: 0px; vertical-align: top;"><div style="line-height: 19.2000007629395px;"> 
</div><div style="line-height: 19.2000007629395px;">For More Detail : <a href="http://conceptbuilders.blogspot.com/2015/07/how-to-find-rank-of-matrix.html" style="color: #007abe; text-decoration: none;">Click Here</a></div><div style="line-height: 19.2000007629395px;"> 
</div></h2></div></div><h2 style="background-color: white; font-stretch: normal; line-height: 20px; margin: 20px 0px 0px; padding: 0px; vertical-align: top;"></h2><h2 style="line-height: 20px; margin: 20px 0px 0px; padding: 0px; vertical-align: top;"></h2></div>

Shortcut To Find Rank of Matrix

Shortcut To Find Rank Of Matrix : Look at the matrix whether it is rectangular or square matrix. For rectangular matrix, if number of rows ...

Basic Concept Of Differentiation

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What is Differentiation?</h2>
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Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.</div>
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What does this mean?</div>
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Constant Rate of Change : </h2>
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First, let's take an example of a car travelling at a constant 60 km/h. The distance-time graph would look like this:</div>
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<img alt="linear graph" height="240" src="http://intmstat.com/differentiation/linear.gif" style="border: none; height: auto; max-width: 95%; padding: 4px;" width="357" /></blockquote>
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 We notice that the distance from the starting point increases at a constant rate of 60 km each hour, so after 5 hours we have travelled 300 km. We notice that the slope (gradient) is always 5300=60 for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)</div>
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Rate of Change that is Not Constant</h2>
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Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive.</div>
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Now let's look at the graph of height (in metres) against time (in seconds).</div>
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<img alt="parabolic curve" height="250" src="http://intmstat.com/differentiation/parab.gif" style="border: none; height: auto; max-width: 95%; padding: 4px;" width="375" /></blockquote>
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Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it becomes 0 (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).</div>
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Tip : The slope of a curve at a point tells us the rate of change of the quantity at that point.</div>
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<img alt="parabola showing slope" height="250" src="http://intmstat.com/differentiation/parab2.gif" style="border: none; height: auto; max-width: 95%; padding: 4px;" width="375" /></blockquote>
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Important Concept - Approximations of the Slope</h3>
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Now, let's zoom in on the section of the graph near t=1 (where I have the rectangle in the graph above). We look at the bit between t = 0.9 s and t = 1.1 s. It looks like this:</div>
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<img alt="parabola zoomed in - slope" height="243" src="http://intmstat.com/differentiation/parab3.gif" style="border: none; height: auto; max-width: 95%; padding: 4px;" width="355" /></blockquote>
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Founders of Calculus</div>
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<img alt="Isaac Newton" height="145" src="http://intmstat.com/applications-integration/Newton.jpg" style="border: none; height: auto; max-width: 95%; padding: 4px;" width="120" /> Sir Isaac Newton <img alt="Gottfried Leibniz" height="145" src="http://intmstat.com/applications-integration/Leibniz.jpg" style="border: none; height: auto; max-width: 95%; padding: 4px;" width="119" /> Gottfried Leibniz</div>
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Notice that if we zoom in close enough to a curve, it begins to look like a straight line. We can find a very good approximation to the slope of the curve at the point t=1 (it will be the slope of the tangent to the curve, marked in dark red) by observing the points that the curve passes through near t=1. (A tangent is a line that touches the curve at one point only.)</div>
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Observing the graph, we see that it passes through (0.9,36.2) and (1.1,42). So the slope of the tangent at t=1 is about:</div>
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slope=x2−x1y2−y1</div>
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=1.1−0.942.0−36.2  =5.8/0.2=29 m/s</blockquote>
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The units are m/s, as this is a velocity. We have found the rate of change by looking at the slope.</div>
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Clearly, if we were to zoom in closer, our curve would look even more straight and we could get an even better approximation for the slope of the curve.</div>
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This idea of "zooming in" on the graph and getting closer and closer to get a better approximation for the slope of the curve (thus giving us the rate of change) was the breakthrough that led to the development of differentiation.</div>
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Development of Differential Calculus</h2>
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Up until the time of Newton and Leibniz, there was no reliable way to describe or predict this constantly changing velocity. There was a real need to understand how constantly varying quantities could be analysed and predicted. That's why they developed differential calculus.</div>
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Why Study Differentiation?</h2>
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There are many applications of differentiation in science and engineering.</div>
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Differentiation is also used in analysis of finance and economics.</div>
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One important application of differentiation is in the area of optimisation, which means finding the condition for a maximum (or minimum) to occur. This is important in business (cost reduction, profit increase) and engineering (maximum strength, minimum cost.)</div>
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Basic Concept Of Differentiation

What is Differentiation? Differentiation is all about finding rates of change of one quantity compared to another. We need differ...

Simple Rules To Find Derivatives of Polynomial

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Common derivatives :</h2>
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A. Derivative of a Constant</h3>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<div class="formula" style="border-radius: 0.4em; border: 1px solid rgb(170, 170, 170); display: inline-block; margin-bottom: 0.5em; padding: 8px 8px 6px 6px;">
dxdc=0</div>
</blockquote>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
This is basic. In English, it means that if a quantity has a constant value, then the rate of change is zero.</div>
<h4 style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px; font-stretch: normal; font-weight: normal; margin: 0px;">
Example : a</h4>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
dxd(6)=0</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
 </h3>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
B. Derivative of n-th power of x</h3>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<div class="formula" style="border-radius: 0.4em; border: 1px solid rgb(170, 170, 170); display: inline-block; margin-bottom: 0.5em; padding: 8px 8px 6px 6px;">
dxdxn=nxn−1</div>
</blockquote>
<h4 style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px; font-stretch: normal; font-weight: normal; margin: 0px;">
Example : b</h4>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
dxdx5=5x4</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
 </h3>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
C. Derivative of Constant product</h3>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<div class="formula" style="border-radius: 0.4em; border: 1px solid rgb(170, 170, 170); display: inline-block; margin-bottom: 0.5em; padding: 8px 8px 6px 6px;">
dxd(cy)=cdxd(y)=cdxdy</div>
</blockquote>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Here, y is some function of x. It means that if we are finding the derivative of a constant times that function, it is the same as finding the derivative of the function first, then multiplying by the constant.</div>
<h4 style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px; font-stretch: normal; font-weight: normal; margin: 0px;">
Example c</h4>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
If y=x7, then dxdy=dxd(x7)=7x6.</div>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Applying the new rule (c), we have:</div>
<div class="alignEquals" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
dxd(3y)=dxd(3x7) <div style="margin-left: 70.9131336212158px;">
=3dxd(x7)</div>
<div style="margin-left: 70.9131336212158px;">
 </div>
<div style="margin-left: 70.9131336212158px;">
=3dxdy</div>
<div style="margin-left: 70.9131336212158px;">
 </div>
<div style="margin-left: 70.9131336212158px;">
=(3)(7x6)</div>
<div style="margin-left: 70.9131336212158px;">
 </div>
<div style="margin-left: 70.9131336212158px;">
=21x6</div>
<div style="margin-left: 70.9131336212158px;">
 </div>
</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
D. Derivative of a sum</h3>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<div class="formula" style="border-radius: 0.4em; border: 1px solid rgb(170, 170, 170); display: inline-block; margin-bottom: 0.5em; padding: 8px 8px 6px 6px;">
dxd(u+v)=dxdu+dxdv</div>
</blockquote>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Here, u and v are functions of x. The derivative of the sum is simply equal to the derivative of the first plus derivative of the second. It does not work the same for the derivative of the product of two functions, that we meet in the next section.</div>
<h4 style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 16px; font-stretch: normal; font-weight: normal; margin: 0px;">
Example : d</h4>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
If u=x2 and v=x9, then:</div>
<div class="alignEquals" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
dxd(u+v)=dxd(x2)+dxd(x9) <div style="margin-left: 76.9346790313721px;">
=2x+9x8</div>
<div style="margin-left: 76.9346790313721px;">
 </div>
</div>
<h2 style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.3em; font-stretch: normal; font-weight: normal; margin: 0px;">
Derivatives Summary</h2>
<table class="tableex" style="background-color: white; border-collapse: collapse; color: black; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;"><tbody>
<tr><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">Constant:</td><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">dxdc=0</td></tr>
<tr><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">n-th power of x:</td><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">dxdxn=nxn−1</td></tr>
<tr><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">Constant product:</td><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">dxd(cy) =cdxd(y)=cdxdy</td></tr>
<tr><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">Sum:</td><td style="border: 1px solid rgb(91, 138, 155); empty-cells: hide; padding: 6px;">dxd(u+v)=dxdu+dxdv</td></tr>
</tbody></table>
<div class="separator" style="background-color: white; clear: both; font-family: Verdana, Geneva, sans-serif; font-size: 12px; line-height: 19.2000007629395px; text-align: center;">
 </div>
<h2 style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.3em; font-stretch: normal; font-weight: normal; margin: 0px;">
Further Examples :</h2>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
Example 1</h3>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Find the derivative of y = −7x6</div>
<div class="answerLink" id="ans0" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; margin-bottom: 2em; margin-top: 1.5em;">
<a class="ajaxtrigger sprite" href="http://conceptbuilders.blogspot.com/" style="background-attachment: initial; background-clip: initial; background-image: url(http://www.intmath.com/includes/images/spritex1opt.png); background-origin: initial; background-position: 0px -645px; background-repeat: no-repeat; background-size: initial; border-radius: 0.5em; border: 1px solid rgb(91, 138, 155); color: #007abe; cursor: url(http://ani.cursors-4u.net/cursors/cur-11/cur1089.cur), progress; margin: 0px; padding: 0.5em 0.7em 0.5em 2em; text-decoration: none;">Answer</a></div>
<div class="ddown" id="target0" style="background-color: white; border-radius: 0.5em; border: 1px solid rgb(91, 138, 155); box-shadow: rgb(170, 170, 170) 0.2em 0.2em 0.2em; clear: both; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; list-style: none; margin: 1em 0px 0px 0.3em; padding: 0.4em;">
Using the rule <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxd(cy)=cdxd(y)</blockquote>
we can take the -7 out the front: <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxd(−7x6)=−7dxd(x6)</blockquote>
And <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxdxn=nxn−1</blockquote>
gives us: <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
−7dxd(x6)=−7×6x5=−42x5</blockquote>
Note: We can do this in one step: <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxdy=−42x5</blockquote>
We can write: dxdy=−42x5 OR y′=−42x5. They mean the same thing.</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
Example 2</h3>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Find the derivative of y = 3x5 − 1</div>
<div class="answerLink" id="ans1" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; margin-bottom: 2em; margin-top: 1.5em;">
<a class="ajaxtrigger sprite" href="http://conceptbuilders.blogspot.com/" style="background-attachment: initial; background-clip: initial; background-image: url(http://www.intmath.com/includes/images/spritex1opt.png); background-origin: initial; background-position: 0px -645px; background-repeat: no-repeat; background-size: initial; border-radius: 0.5em; border: 1px solid rgb(91, 138, 155); color: #007abe; cursor: url(http://ani.cursors-4u.net/cursors/cur-11/cur1089.cur), progress; margin: 0px; padding: 0.5em 0.7em 0.5em 2em; text-decoration: none;">Answer</a></div>
<div class="ddown" id="target1" style="background-color: white; border-radius: 0.5em; border: 1px solid rgb(91, 138, 155); box-shadow: rgb(170, 170, 170) 0.2em 0.2em 0.2em; clear: both; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; list-style: none; margin: 1em 0px 0px 0.3em; padding: 0.4em;">
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y=3x5−1</blockquote>
Now, <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxd(3x5)=3×5x4=15x4</blockquote>
And since dxdc=0, we can write: <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxd(−1)=0</blockquote>
So <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxdy=dxd(3x5−1)=15x4</blockquote>
</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
Example 3</h3>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Find the derivative of</div>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y=13x4−6x3−x−1</blockquote>
<div class="answerLink" id="ans2" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; margin-bottom: 2em; margin-top: 1.5em;">
<a class="ajaxtrigger sprite" href="http://conceptbuilders.blogspot.com/" style="background-attachment: initial; background-clip: initial; background-image: url(http://www.intmath.com/includes/images/spritex1opt.png); background-origin: initial; background-position: 0px -645px; background-repeat: no-repeat; background-size: initial; border-radius: 0.5em; border: 1px solid rgb(91, 138, 155); color: #007abe; cursor: url(http://ani.cursors-4u.net/cursors/cur-11/cur1089.cur), progress; margin: 0px; padding: 0.5em 0.7em 0.5em 2em; text-decoration: none;">Answer</a></div>
<div class="ddown" id="target2" style="background-color: white; border-radius: 0.5em; border: 1px solid rgb(91, 138, 155); box-shadow: rgb(170, 170, 170) 0.2em 0.2em 0.2em; clear: both; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; list-style: none; margin: 1em 0px 0px 0.3em; padding: 0.4em;">
Now, taking each term in turn: dxd(13x4)=52x3 (using dxdxn=nxn−1) dxd(−6x3)=−18x2 (using dxdxn=nxn−1) dxd(−x)=−1 (since −x=−(x1) and so the derivative will be −(x0)=−1) dxd(−1)=0 (since dxdc=0) So <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxdy=52x3−18x2−1</blockquote>
</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
Example 4</h3>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Find the derivative of</div>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y=−41x8+21x4−32</blockquote>
<div class="answerLink" id="ans3" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; margin-bottom: 2em; margin-top: 1.5em;">
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<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y=−41x8+21x4−32</blockquote>
Differentiating term by term, we have: <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxd(−41x8)=−48x7=−2x7</blockquote>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxd(21x4)=24x3=2x3 dxd(32)=0 (this is the derivative ofa constant)</blockquote>
So dxdy=dxd(−41x8+21x4−32)=−2x7+2x3</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
Example 5</h3>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Evaluate the derivative of</div>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y=x4−9x2−5x</blockquote>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
at the point (3,15).</div>
<div class="answerLink" id="ans4" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; margin-bottom: 2em; margin-top: 1.5em;">
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y=x4−9x2−5x So <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxdy=4x3−18x−5</blockquote>
At the point where x=3, the derivative has value: <blockquote class="alignEquals" style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxdy=4(3)3−18(3)−5 <div style="margin-left: 32.7784290313721px;">
=4×27−18×3−5</div>
<div style="margin-left: 32.7784290313721px;">
=49</div>
</blockquote>
This means that the slope of the curve y=x4−9x2−5x at x=3 is 49.</div>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.1em; font-stretch: normal; font-weight: normal; margin: 0px;">
Example 6</h3>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Find the derivative of the function</div>
<blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat rgb(255, 255, 255); color: #777777; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y=x1/4−x2</blockquote>
<div class="answerLink" id="ans5" style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px; margin-bottom: 2em; margin-top: 1.5em;">
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In this case we have fractions and negative numbers for the powers of x. (So it is not a polynomial). The differentiation rules still apply. y=x1/4−x2 We can write this as: y=x1/4−2x−1 Differentiating gives us: <blockquote class="alignEquals" style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
dxdy=41x41−1−2(−1)x−1−1 <div style="margin-left: 32.7784290313721px;">
=41x−3/4+2x−2</div>
<div style="margin-left: 32.7784290313721px;">
=4x3/41+x22</div>
</blockquote>
</div>
<h2 style="background-color: white; font-family: Georgia, Times, 'Times New Roman', serif; font-size: 1.3em; font-stretch: normal; font-weight: normal; margin: 0px;">
Exercise</h2>
<div style="background-color: white; font-family: Arial, Helvetica, sans-serif; font-size: 12px; line-height: 19.2000007629395px;">
Find the equation of the tangent to the curve y=3x−x3 at x=2.</div>
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Now y=3x−x3 dxdy=3−3x2 and the value of this derivative at x=2 is given by: dxdy=3−3(2)2=−9 Since y=3x−x3, then when x=2, y=−2. So we need the equation of the line passing through (2,−2) with slope −9. Using the general equation of the line y−y1=m(x−x1), we have: <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y+2=−9(x−2)</blockquote>
So the required equation is: <blockquote style="background: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png) 0% 5px no-repeat; color: #777777; font-size: 13px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
y=−9x+16</blockquote>
Or, in general form: 9x+y−16=0.</div>
</div>

Simple Rules To Find Derivatives of Polynomial

Common derivatives : A. Derivative of a Constant d x d c = 0 This is basic. In English, it means that if a quantity ...