009A Sample Midterm 3, Problem 6

Find the derivatives of the following functions. Do not simplify.

a)$f(x)=\sin {\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}$

b)$g(x)={\sqrt {\frac {x^{2}+2}{x^{2}+4}}}$

c)$h(x)=(x+\cos ^{2}x)^{8}$

Foundations:
1. Chain Rule
2. Quotient Rule

Solution:

(a)

Step 1:
First, using the Chain Rule, we have
$f'(x)=\cos {\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}{\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}'.$

Step 2:
Now, using the Quotient Rule and Chain Rule, we have
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {\cos {\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}{\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}'}\\&&\\&=&\displaystyle {\cos {\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}{\bigg (}{\frac {e^{-x}(x^{-3})'-x^{-3}(e^{-x})'}{(e^{-x})^{2}}}{\bigg )}}\\&&\\&=&\displaystyle {\cos {\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}{\bigg (}{\frac {e^{-x}(-3x^{-4})-x^{-3}(e^{-x})(-x)'}{(e^{-x})^{2}}}{\bigg )}}\\&&\\&=&\displaystyle {\cos {\bigg (}{\frac {x^{-3}}{e^{-x}}}{\bigg )}{\bigg (}{\frac {e^{-x}(-3x^{-4})-x^{-3}(e^{-x})(-1)}{(e^{-x})^{2}}}{\bigg )}.}\end{array}}$

(b)

Step 1:
First, using the Chain Rule, we have
$g'(x)={\frac {1}{2}}{\bigg (}{\frac {x^{2}+2}{x^{2}+4}}{\bigg )}^{-{\frac {1}{2}}}{\bigg (}{\frac {x^{2}+2}{x^{2}+4}}{\bigg )}'.$

Step 2:
Now, using the Quotient Rule, we have
${\begin{array}{rcl}\displaystyle {g'(x)}&=&\displaystyle {{\frac {1}{2}}{\bigg (}{\frac {x^{2}+2}{x^{2}+4}}{\bigg )}^{-{\frac {1}{2}}}{\bigg (}{\frac {x^{2}+2}{x^{2}+4}}{\bigg )}'}\\&&\\&=&\displaystyle {{\frac {1}{2}}{\bigg (}{\frac {x^{2}+2}{x^{2}+4}}{\bigg )}^{-{\frac {1}{2}}}{\bigg (}{\frac {(x^{2}+4)(x^{2}+2)'-(x^{2}+2)(x^{2}+4)'}{(x^{2}+4)^{2}}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {1}{2}}{\bigg (}{\frac {x^{2}+2}{x^{2}+4}}{\bigg )}^{-{\frac {1}{2}}}{\bigg (}{\frac {(x^{2}+4)(2x)-(x^{2}+2)(2x)}{(x^{2}+4)^{2}}}{\bigg )}.}\end{array}}$

(c)

Step 1:
First, using the Chain Rule, we have
$h'(x)=8(x+\cos ^{2}(x))^{7}(x+\cos ^{2}(x))'.$

Step 2:
Now, using the Chain Rule again we get
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} \displaystyle{h'(x)} & = & \displaystyle{8(x+\cos^2(x))^7(x+\cos^2(x))'}\\ &&\\ & = & \displaystyle{8(x+\cos^2(x))^7(1+2\cos(x)(\cos(x))')}\\ &&\\ & = & \displaystyle{8(x+\cos^2(x))^7(1-2\cos(x)\sin(x)).} \end{array}}

Final Answer:
(a)     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 \cos\bigg(\frac{x^{-3}}{e^{-x}}\bigg)\bigg(\frac{e^{-x}(-3x^{-4})-x^{-3}(e^{-x})(-1)}{(e^{-x})^2}\bigg)}
(b)     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{1}{2}\bigg(\frac{x^2+2}{x^2+4}\bigg)^{-\frac{1}{2}}\bigg(\frac{(x^2+4)(2x)-(x^2+2)(2x)}{(x^2+4)^2}\bigg)}
(c)     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 8(x+\cos^2(x))^7(1-2\cos(x)\sin(x))}

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