MathAdmin at 00:22, 7 June 2015

2015-06-07T00:22:38Z

MathAdmin: Created page with "Find the derivative: $g(x) = \frac{ln(x^3 + 7)}{(x^4 + 2x^2)}$ . ''(Note: You do not ne..."

2015-06-05T03:44:46Z

Created page with "Find the derivative: <math style="vertical-align: -18px">g(x) = \frac{ln(x^3 + 7)}{(x^4 + 2x^2)}</math> . ''(Note: You do not ne..."

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Find the derivative: <math style="vertical-align: -18px">g(x) = \frac{ln(x^3 + 7)}{(x^4 + 2x^2)}</math> .

''(Note: You do not need to simplify the derivative after finding it.)''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|This problem requires some more advanced rules of differentiation. In particular, it needs
|-
|'''The Chain Rule:''' If <math style="vertical-align: -25%;">f</math> and <math style="vertical-align: -15%;">g</math> are differentiable functions, then
|-

|      <math>(f\circ g)'(x) = f'(g(x))\cdot g'(x).</math>
|-
| '''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> , then
|-
|      <math>\left(\frac{f}{g}\right)'(x) = \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^2}. </math>
|-
| 
|}

 '''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Find the derivative of the denominator:  
|-
|We need to use the chain rule, where the inner function is <math>x^3 + 7</math> and the outer function is natural log:
|-
|
::<math>\begin{array}{rcl}
\left[\ln(x^{3}+7)\right]' & = & {\displaystyle \frac{1}{x^{3}+7}\cdot3x^{2}}\\
\\
& = & {\displaystyle \frac{3x^{2}}{x^{3}+7}.}
\end{array}</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Apply the Quotient Rule:  
|-
|
::<math>\begin{array}{rcl}
\left[{\displaystyle \frac{\ln(x^{3}+7)}{x^{4}+2x^{2}}} \right]' & = & {\displaystyle \frac{\left[\ln(x^{3}+7)\right]'\cdot\left(x^{4}+2x^{2}\right)-\left(x^{4}+2x^{2}\right)'\cdot\ln(x^{3}+7)}{\left(x^{4}+2x^{2}\right)^{2}}}\\
\\
& = & {\displaystyle \frac{\frac{3x^{2}}{x^{3}+7}\cdot\left(x^{4}+2x^{2}\right)-\left(4x^{3}+4x\right)\cdot\ln(x^{3}+7)}{\left(x^{4}+2x^{2}\right)^{2}}}.\\
\\
\end{array}</math>

|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|
::<math>\left[\frac{\ln(x^{3}+7)}{x^{4}+2x^{2}}\right]'\,=\,\frac{\frac{3x^{2}}{x^{3}+7}\cdot\left(x^{4}+2x^{2}\right)-\left(4x^{3}+4x\right)\cdot\ln(x^{3}+7)}{\left(x^{4}+2x^{2}\right)^{2}}.</math>
|}

[[022_Sample_Final_A|'''Return to Sample Exam''']]

@@ Line 25: / Line 25: @@
 !Find the derivative of the denominator: &nbsp;
 |-
-|We need to use the chain rule, where the inner function is <math>x^3 + 7</math> and the outer function is natural log:
+|We need to use the chain rule, where the inner function is <math style="vertical-align: -2px">x^3 + 7</math> and the outer function is natural log:
 |-
+|

022 Sample Final A, Problem 11 - Revision history

MathAdmin at 00:22, 7 June 2015

MathAdmin: Created page with "Find the derivative: g(x) = \frac{ln(x^3 + 7)}{(x^4 + 2x^2)} . ''(Note: You do not ne..."

MathAdmin: Created page with "Find the derivative: $g(x) = \frac{ln(x^3 + 7)}{(x^4 + 2x^2)}$ . ''(Note: You do not ne..."