009A Sample Midterm 2, Problem 5

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

a) $f(x)=\tan ^{3}(7x^{2}+5)$

b) $g(x)=\sin(\cos(e^{x}))$

c) $h(x)={\frac {(5x^{2}+7x)^{3}}{\ln(x^{2}+1)}}$

ExpandFoundations:
1. Chain Rule
2. Derivatives of trig/ln
3. Quotient Rule

Solution:

(a)

ExpandStep 1:
First, we use the Chain Rule to get
$f'(x)=3\tan ^{2}(7x^{2}+5)(\tan(7x^{2}+5))'.$

ExpandStep 2:
Now, we use the Chain Rule again to get
${\begin{array}{rcl}\displaystyle {f'(x)}&=&\displaystyle {3\tan ^{2}(7x^{2}+5)(\tan(7x^{2}+5))'}\\&&\\&=&\displaystyle {3\tan ^{2}(7x^{2}+5)\sec ^{2}(7x^{2}+5)(7x^{2}+5)'}\\&&\\&=&\displaystyle {3\tan ^{2}(7x^{2}+5)\sec ^{2}(7x^{2}+5)(14x).}\end{array}}$

(b)

ExpandStep 1:
First, we use the Chain Rule to get
$g'(x)=\cos(\cos(e^{x}))(\cos(e^{x}))'.$

ExpandStep 2:
Now, we use the Chain Rule again to get
${\begin{array}{rcl}\displaystyle {g'(x)}&=&\displaystyle {\cos(\cos(e^{x}))(\cos(e^{x}))'}\\&&\\&=&\displaystyle {\cos(\cos(e^{x}))(-\sin(e^{x}))(e^{x})'}\\&&\\&=&\displaystyle {\cos(\cos(e^{x}))(-\sin(e^{x}))(e^{x}).}\end{array}}$

(c)

ExpandStep 1:

ExpandStep 2:

ExpandFinal Answer:
(a)     $3\tan ^{2}(7x^{2}+5)\sec ^{2}(7x^{2}+5)(14x)$
(b)     $\cos(\cos(e^{x}))(-\sin(e^{x}))(e^{x})$
(c)

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