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

Basic Concept Of Differentiation

<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixP3u2gwvNlzGSJrG-EWYKTj6TW7d5FJcZR_zw0xL5uVEht3c-kuX05MFI6f2mkDlr6xkrYH8DY3934oIUnwwVTSURt17ZcmmLAWw4RHtxkp58-b2D3mox2iBpkfVzF4Xs919KfhT-Ihs/s1600/equations.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixP3u2gwvNlzGSJrG-EWYKTj6TW7d5FJcZR_zw0xL5uVEht3c-kuX05MFI6f2mkDlr6xkrYH8DY3934oIUnwwVTSURt17ZcmmLAWw4RHtxkp58-b2D3mox2iBpkfVzF4Xs919KfhT-Ihs/s320/equations.jpg" width="320" /></a></div>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; margin: 0px;">
 </h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; margin: 0px;">
What is Differentiation?</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
<div style="line-height: 19.2000007629395px;">
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>
<div style="line-height: 19.2000007629395px;">
What does this mean?</div>
</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; margin: 0px;">
Constant Rate of Change : </h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
<div style="line-height: 19.2000007629395px;">
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>
<blockquote style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png); background-origin: initial; background-position: 0% 5px; background-repeat: no-repeat; background-size: initial; color: #777777; font-size: 21px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<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>
<div style="line-height: 19.2000007629395px;">
 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>
</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
Rate of Change that is Not Constant</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
<div style="line-height: 19.2000007629395px;">
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>
<div style="line-height: 19.2000007629395px;">
Now let's look at the graph of height (in metres) against time (in seconds).</div>
<blockquote style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png); background-origin: initial; background-position: 0% 5px; background-repeat: no-repeat; background-size: initial; color: #777777; font-size: 21px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<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>
<div style="line-height: 19.2000007629395px;">
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>
</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; margin: 0px;">
<div class="dunmiss" style="background-attachment: initial; background-clip: initial; background-color: #e5edef; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-radius: 4px; border: 1px solid rgb(74, 121, 138); float: right; line-height: 19.2000007629395px; margin: 0px 0px 15px 20px; padding: 5px; width: 190px;">
Tip : The slope of a curve at a point tells us the rate of change of the quantity at that point.</div>
<blockquote style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png); background-origin: initial; background-position: 0% 5px; background-repeat: no-repeat; background-size: initial; color: #777777; font-size: 21px; font-weight: normal; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<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>
</h2>
<h3 style="background: rgb(255, 255, 255); font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 17px; font-stretch: normal; font-weight: normal; margin: 0px;">
Important Concept - Approximations of the Slope</h3>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
<div style="line-height: 19.2000007629395px;">
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>
<blockquote style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png); background-origin: initial; background-position: 0% 5px; background-repeat: no-repeat; background-size: initial; color: #777777; font-size: 21px; line-height: 1.6em; margin: 20px; padding-left: 40px;">
<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>
</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; margin: 0px;">
<div class="dunmiss" style="background-attachment: initial; background-clip: initial; background-color: #e5edef; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border-radius: 4px; border: 1px solid rgb(74, 121, 138); float: right; line-height: 19.2000007629395px; margin: 0px 0px 15px 20px; padding: 5px; width: 190px;">
<div style="text-align: center;">
Founders of Calculus</div>
<div style="font-weight: normal; text-align: center;">
<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>
</div>
<div style="font-weight: normal; line-height: 19.2000007629395px;">
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>
<div style="font-weight: normal; line-height: 19.2000007629395px;">
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>
<div style="font-weight: normal; line-height: 19.2000007629395px;">
slope=x2−x1y2−y1</div>
<blockquote style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png); background-origin: initial; background-position: 0% 5px; background-repeat: no-repeat; background-size: initial; color: #777777; font-size: 21px; font-weight: normal; line-height: 1.6em; margin: 20px; padding-left: 40px;">
=1.1−0.942.0−36.2  =5.8/0.2=29 m/s</blockquote>
<blockquote style="background-attachment: initial; background-clip: initial; background-color: initial; background-image: url(https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPdVAsZ8vKJ4zgeCP4oArf1LSKfdsRRtjr9R9HmON16uRmey0s6sYfZSvSVbADO54kJUXNQfydPMKakIdWAlTVYCwpcx_j5NrTEBwIf6BE_OlRtfKboepjC90-ecXssQJ0ZnpwCr59vilq/s1600/quotation-l.png); background-origin: initial; background-position: 0% 5px; background-repeat: no-repeat; background-size: initial; color: #777777; font-size: 21px; font-weight: normal; line-height: 1.6em; margin: 20px; padding-left: 40px;">
</blockquote>
<div style="font-weight: normal; line-height: 19.2000007629395px;">
The units are m/s, as this is a velocity. We have found the rate of change by looking at the slope.</div>
<div style="font-weight: normal; line-height: 19.2000007629395px;">
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>
<div style="font-weight: normal; line-height: 19.2000007629395px;">
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>
</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
Development of Differential Calculus</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
<div style="line-height: 19.2000007629395px;">
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>
</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
Why Study Differentiation?</h2>
<h2 style="background-color: white; font-family: Georgia, Utopia, 'Palatino Linotype', Palatino, serif; font-size: 19px; font-stretch: normal; font-weight: normal; margin: 0px;">
<div style="line-height: 19.2000007629395px;">
There are many applications of differentiation in science and engineering.</div>
<div style="line-height: 19.2000007629395px;">
Differentiation is also used in analysis of finance and economics.</div>
<div style="line-height: 19.2000007629395px;">
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>
</h2>
<h2 style="background-color: white;">
</h2>
</div>

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

<div dir="ltr" style="text-align: left;" trbidi="on">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://www.blogger.com/blogger.g?blogID=1267300818045385815" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWjV_VJubsu3lB0ecssFXZIqMxcozvepcZswIzzo1ME0jwcQ85G3_XGIJmnF0_S_Uenr5ewApc7IDYLgfc2JNnHERVKm4sdZJcg6pG9plpZUpbDnwa7iYcq92hCBNXT08xeE42ZVj9hCo/s1600/math_arithmetic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="141" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWjV_VJubsu3lB0ecssFXZIqMxcozvepcZswIzzo1ME0jwcQ85G3_XGIJmnF0_S_Uenr5ewApc7IDYLgfc2JNnHERVKm4sdZJcg6pG9plpZUpbDnwa7iYcq92hCBNXT08xeE42ZVj9hCo/s200/math_arithmetic.jpg" width="200" /></a></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;">
 </h2>
<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;">
Common derivatives :</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;">
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;">
<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="target3" 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=−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;">
<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="target4" 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;">
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;">
<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="target5" 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;">
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>
<div class="answerLink" id="ans6" 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="target6" 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 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 ...