Difference between revisions of "022 Exam 2 Sample A, Problem 1"

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!Foundations:    
 
!Foundations:    
 
|-
 
|-
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|This problem requires several advanced rules of differentiation.  In particular, you need
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|This problem is best approached through properties of logarithms.  Remember that
 
|-
 
|-
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|'''The Chain Rule:''' If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -15%;">g</math> are differentiable functions, then
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|<br>&nbsp;&nbsp;&nbsp;&nbsp; <math>\ln (xy) = \ln x + \ln y,</math>
 
|-
 
|-
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|and
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|<br>&nbsp;&nbsp;&nbsp;&nbsp; <math>(f\circ g)'(x) = f'(g(x))\cdot g'(x).</math>
 
 
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|-
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|<br>'''The Product Rule:'''  If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -15%;">g</math> are differentiable functions, then
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|<br>&nbsp;&nbsp;&nbsp;&nbsp; <math>\ln \left( \frac{x}{y}\right) = \ln x - \ln y,</math>
 
|-
 
|-
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|<br>&nbsp;&nbsp;&nbsp;&nbsp; <math>(fg)'(x) = f'(x)\cdot g(x)+f(x)\cdot g'(x).</math>
 
 
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|-
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|<br>'''The Quotient Rule:'''  If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -15%;">g</math> are differentiable functions and <math style="vertical-align: -21%;">g(x) \neq 0</math>&thinsp;, then
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|You will also need to apply
 
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|-
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|<br>&nbsp;&nbsp;&nbsp;&nbsp; <math>\left(\frac{f}{g}\right)'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^2}. </math>
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|'''The Chain Rule:''' If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -15%;">g</math> are differentiable functions, then
 
|-
 
|-
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|Additionally, we will need our power rule for differentiation:
+
 
 +
|<br>&nbsp;&nbsp;&nbsp;&nbsp; <math>(f\circ g)'(x) = f'(g(x))\cdot g'(x).</math>
 
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|-
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|
+
 
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::<math style="vertical-align: -21%;">\left(x^n\right)'\,=\,nx^{n-1},</math> for <math style="vertical-align: -25%;">n\neq 0</math>,
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|Finally, recall that the derivative of natural log is
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|-
 
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|as well as the derivative of natural log:
 
 
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|-
 
|
 
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!Step 1: &nbsp;
 
!Step 1: &nbsp;
 
|-
 
|-
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|We need to identify the composed functions in order to apply the chain rule. Note that if we set <math style="vertical-align: -21%">g(x)\,=\,\ln x</math>, and
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|We can use the log rules to rewrite our function as  
 
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|-
 
|
 
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::<math>f(x)\,=\,\frac{(x+5)(x-1)}{x},</math>
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::<math>y\,=\,\ln (x+5)+\ln(x-1)-\ln(x).</math>
 
|-
 
|-
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|we then have&thinsp; <math style="vertical-align: -21%">y\,=\,g\circ f(x)\,=\,g\left(f(x)\right).</math>
 
 
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!Step 2: &nbsp;
 
!Step 2: &nbsp;
 
|-
 
|-
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|We can now apply the advanced techniques.  For <math style="vertical-align: -20%">f'(x)</math>, we can avoid using the product rule by first multiplying out the denominator.  Then by the quotient rule,
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|We can differentiate term-by-term, applying the chain rule to the first two to find
 
|-
 
|-
 
|<br>
 
|<br>
 
::<math>\begin{array}{rcl}
 
::<math>\begin{array}{rcl}
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f'(x) & = & \displaystyle{\frac{\left[(x+5)(x-1)\right]'x-(x+5)(x-1)(x)'}{x^{2}}}\\
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y' & = & \displaystyle{\frac{1}{x+5}\cdot(x+5)'+\frac{1}{x-1}\cdot(x-1)'+\frac{1}{x}}\\
 
\\
 
\\
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  & = &  \displaystyle{\frac{\left(x^{2}+4x-5\right)'x-(x^{2}+4x-5)(x)'}{x^{2}}}\\
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  & = &  \displaystyle{\frac{1}{x+5}+\frac{1}{x-1}+\frac{1}{x}}.
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\\
 
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& = &  \displaystyle{\frac{(2x+4)x-(x^{2}+4x-5)(1)}{x^{2}}}\\
 
−
\\
 
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& = &  \displaystyle{\frac{2x^{2}+4x-x^{2}-4x+5}{x^{2}}}\\
 
−
\\
 
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& = &  \displaystyle{\frac{x^{2}+5}{x^{2}}}.
 
−
\end{array}</math>
 
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|
 
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|}
 
  
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
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!Step 3: &nbsp;
 
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|-
 
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|We can now use the chain rule to find<br>
 
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|-
 
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|
 
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::<math>\begin{array}{rcl}
 
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y' & = & \left(g\circ f\right)'(x)\\
 
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\\
 
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& = & g'\left(f(x)\right)\cdot f'(x)\\
 
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\\& = & \displaystyle{\frac{x}{(x+5)(x-1)}\cdot\frac{x^{2}+5}{x^{2}}}\\
 
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\\
 
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& = & \displaystyle{\frac{x^{2}+5}{x^{3}+4x^{2}-5x}.}
 
 
\end{array}</math>
 
\end{array}</math>
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Note that many teachers <u>'''do not'''</u> prefer a cleaned up answer, and may request that you <u>'''do not simplify'''</u>.  In this case, we could write the answer as<br>
 
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|-
 
 
|
 
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::<math>y'=\displaystyle {\frac{x}{(x+5)(x-1)}\cdot\frac{(2x+4)x-(x^{2}+4x-5)(1)}{x^{2}}.}</math>
 
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|}
 
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!Final Answer: &nbsp;
 
!Final Answer: &nbsp;
 
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|-
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|<math>y'\,=\,\displaystyle{\frac{x^{2}+5}{x^{3}+4x^{2}-5x}.}</math>
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|<math>y'\,=\,\displaystyle{\frac{1}{x+5}+\frac{1}{x-1}+\frac{1}{x}}.</math>
 
|}
 
|}
  
 
[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]
 
[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

Revision as of 21:40, 18 January 2017

Find the derivative of  

Foundations:  
This problem is best approached through properties of logarithms. Remember that

    
and

    
You will also need to apply
The Chain Rule: If and are differentiable functions, then

    
Finally, recall that the derivative of natural log is

 Solution:

Step 1:  
We can use the log rules to rewrite our function as
Step 2:  
We can differentiate term-by-term, applying the chain rule to the first two to find

Final Answer:  

Return to Sample Exam

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