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:

Step 2:

(b)

Step 1:

Step 2:

(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
${\begin{array}{rcl}\displaystyle {h'(x)}&=&\displaystyle {8(x+\cos ^{2}(x))^{7}(x+\cos ^{2}(x))'}\\&&\\&=&\displaystyle {8(x+\cos ^{2}(x))^{7}(1+2\cos(x)(\cos(x))')}\\&&\\&=&\displaystyle {8(x+\cos ^{2}(x))^{7}(1-2\cos(x)\sin(x)).}\end{array}}$

Final Answer:
(a)
(b)
(c)     $8(x+\cos ^{2}(x))^{7}(1-2\cos(x)\sin(x))$

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