# 009A Sample Midterm 2, Problem 5 Detailed Solution

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)   $h(x)={\frac {(5x^{2}+7x)^{3}}{\ln(x^{2}+1)}}$ Background Information:
1. Chain Rule
${\frac {d}{dx}}(f(g(x)))=f'(g(x))g'(x)$ 2. Trig Derivatives
${\frac {d}{dx}}(\sin x)=\cos x,\quad {\frac {d}{dx}}(\cos x)=-\sin x$ 3. Quotient Rule
${\frac {d}{dx}}{\bigg (}{\frac {f(x)}{g(x)}}{\bigg )}={\frac {g(x)f'(x)-f(x)g'(x)}{(g(x))^{2}}}$ 4. Derivative of natural logarithm
${\frac {d}{dx}}(\ln x)={\frac {1}{x}}$ Solution:

(a)

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

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

Step 1:
First, we use the Quotient Rule to get
$h'(x)={\frac {\ln(x^{2}+1)((5x^{2}+7x)^{3})'-(5x^{2}+7x)^{3}(\ln(x^{2}+1))'}{(\ln(x^{2}+1))^{2}}}.$ Step 2:
Now, we use the Chain Rule to get
${\begin{array}{rcl}\displaystyle {h'(x)}&=&\displaystyle {\frac {\ln(x^{2}+1)((5x^{2}+7x)^{3})'-(5x^{2}+7x)^{3}(\ln(x^{2}+1))'}{(\ln(x^{2}+1))^{2}}}\\&&\\&=&\displaystyle {\frac {\ln(x^{2}+1)3(5x^{2}+7x)^{2}(5x^{2}+7x)'-(5x^{2}+7x)^{3}{\frac {1}{x^{2}+1}}(x^{2}+1)'}{(\ln(x^{2}+1))^{2}}}\\&&\\&=&\displaystyle {{\frac {\ln(x^{2}+1)3(5x^{2}+7x)^{2}(10x+7)-(5x^{2}+7x)^{3}{\frac {1}{x^{2}+1}}(2x)}{(\ln(x^{2}+1))^{2}}}.}\end{array}}$ (a)     $f'(x)=3\tan ^{2}(7x^{2}+5)\sec ^{2}(7x^{2}+5)(14x)$ (b)     $g'(x)=\cos(\cos(e^{x}))(-\sin(e^{x}))(e^{x})$ (c)     $h'(x)={\frac {\ln(x^{2}+1)3(5x^{2}+7x)^{2}(10x+7)-(5x^{2}+7x)^{3}{\frac {1}{x^{2}+1}}(2x)}{(\ln(x^{2}+1))^{2}}}$ 