022 Sample Final A, Problem 11

Find the derivative: 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) = \frac{ln(x^3 + 7)}{(x^4 + 2x^2)}}  .

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

Foundations:
This problem requires some more advanced rules of differentiation. In particular, it needs
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
$\left({\frac {f}{g}}\right)'(x)={\frac {f'(x)\cdot g(x)-f(x)\cdot g'(x)}{\left(g(x)\right)^{2}}}.$

 Solution:

Find the derivative of the denominator:
We need to use the chain rule, where the inner function is $x^{3}+7$ and the outer function is natural log:
${\begin{array}{rcl}\left[\ln(x^{3}+7)\right]'&=&{\displaystyle {\frac {1}{x^{3}+7}}\cdot 3x^{2}}\\\\&=&{\displaystyle {\frac {3x^{2}}{x^{3}+7}}.}\end{array}}$

Apply the Quotient Rule:

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 \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}}

Final Answer:

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{\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}}.}

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