Difference between revisions of "022 Exam 1 Sample A, Problem 8"

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<span style="font-size:135%">8. Find the derivative of the function <math style="vertical-align: -43%">f(x)=\frac{(3x-1)^{2}}{x^{3}-7}</math>.
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<span style="font-size:135%"><font face=Times Roman>8. Find the derivative of the function <math style="vertical-align: -43%">f(x)=\frac{(3x-1)^{2}}{x^{3}-7}</math>.
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You do not need to simplify your answer.
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You do not need to simplify your answer.</font face=Times Roman>
  
 
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&nbsp;'''Solution:'''
 
  
 
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!Step 1: &nbsp;
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!Solution: &nbsp;
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|Note that we need to use chain rule to find the derivative of <math style="vertical-align: -25%;">\left(3x-1\right)^2</math>.  Then we find
 
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    <td> &nbsp;&nbsp;&nbsp;&nbsp; <math style="vertical-align: -70%">f'(x)</math>&nbsp; </td>
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    <td><math>=\,\,\frac{\left[\left(3x-1\right)^{2}\right]'\cdot(x^{3}-7) \,\,-\,\, \left(3x-1\right)^{2}\cdot(x^{3}-7)'}{(x^{3}-7)^{2}}</math></td>
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    <td><math>=\frac{\left[2\left(3x-1\right)\cdot3\right]\cdot(x^{3}-7) \,\,-\,\, \left(3x-1\right)^{2}\cdot3x^{2}}{(x^{3}-7)^{2}}.</math></td>
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[[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]
 
[[022_Exam_1_Sample_A|'''<u>Return to Sample Exam</u>''']]

Revision as of 14:59, 2 April 2015

8. Find the derivative of the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{(3x-1)^{2}}{x^{3}-7}} . You do not need to simplify your answer.

Foundations:  
This problem involves some more advanced rules of differentiation. In particular, it requires
The Chain Rule: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are differentiable functions, then

     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f\circ g)'(x) = f'(g(x))\cdot g'(x).}

The Quotient Rule: If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are differentiable functions and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) \neq 0}  , then

     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{f}{g}\right)'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^2}. }

Solution:  
Note that we need to use chain rule to find the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(3x-1\right)^2} . Then we find
     Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)}   Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\,\,\frac{\left[\left(3x-1\right)^{2}\right]'\cdot(x^{3}-7) \,\,-\,\, \left(3x-1\right)^{2}\cdot(x^{3}-7)'}{(x^{3}-7)^{2}}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{\left[2\left(3x-1\right)\cdot3\right]\cdot(x^{3}-7) \,\,-\,\, \left(3x-1\right)^{2}\cdot3x^{2}}{(x^{3}-7)^{2}}.}


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