007A Sample Final 3, Problem 2 Detailed Solution

Find the derivative of the following functions:

(a)  $g(\theta )={\frac {\pi ^{2}}{(\sec \theta -\sin 2\theta )^{2}}}$

(b)  $y=\cos(3\pi )+\tan ^{-1}({\sqrt {x}})$

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Background Information:
1. Chain Rule
${\frac {d}{dx}}(f(g(x)))=f'(g(x))g'(x)$
2. Trig Derivatives
${\frac {d}{dx}}(\sin x)=\cos x,\quad {\frac {d}{dx}}(\sec x)=\sec x\tan x$
3. Inverse Trig Derivatives
${\frac {d}{dx}}(\tan ^{-1}x)={\frac {1}{1+x^{2}}}$

Solution:

(a)

Step 1:
First, we write
$g(\theta )=\pi ^{2}(\sec \theta -\sin 2\theta )^{-2}.$
Now, using the Chain Rule, we have
$g'(\theta )=(-2)\pi ^{2}(\sec \theta -\sin 2\theta )^{-3}\cdot {\frac {d}{d\theta }}(\sec \theta -\sin 2\theta ).$

Step 2:
Now, using the Chain Rule a second time, we get
${\begin{array}{rcl}\displaystyle {g'(\theta )}&=&\displaystyle {(-2)\pi ^{2}(\sec \theta -\sin 2\theta )^{-3}\cdot {\frac {d}{d\theta }}(\sec \theta -\sin 2\theta )}\\&&\\&=&\displaystyle {(-2)\pi ^{2}(\sec \theta -\sin 2\theta )^{-3}{\bigg (}\sec \theta \tan \theta -\cos(2\theta )\cdot {\frac {d}{d\theta }}(2\theta ){\bigg )}}\\&&\\&=&\displaystyle {(-2)\pi ^{2}(\sec \theta -\sin 2\theta )^{-3}(\sec \theta \tan \theta -\cos(2\theta )(2))}\\&&\\&=&\displaystyle {{\frac {-2\pi ^{2}(\sec \theta \tan \theta -2\cos(2\theta ))}{(\sec \theta -\sin 2\theta )^{3}}}.}\end{array}}$

(b)

Step 1:
First, we have
$y'={\frac {d}{dx}}(\cos(3\pi ))+{\frac {d}{dx}}(\tan ^{-1}({\sqrt {x}})).$
Since  $\cos(3\pi )$  is a constant,
we have
${\frac {d}{dx}}(\cos(3\pi ))=0.$
Therefore,
$y'={\frac {d}{dx}}(\tan ^{-1}({\sqrt {x}})).$

Step 2:
Now, using the Chain Rule, we have
${\begin{array}{rcl}\displaystyle {y'}&=&\displaystyle {{\frac {d}{dx}}(\tan ^{-1}({\sqrt {x}}))}\\&&\\&=&\displaystyle {{\bigg (}{\frac {1}{1+({\sqrt {x}})^{2}}}{\bigg )}\cdot {\frac {d}{dx}}({\sqrt {x}})}\\&&\\&=&\displaystyle {{\bigg (}{\frac {1}{1+x}}{\bigg )}{\frac {1}{2{\sqrt {x}}}}}\\&&\\&=&\displaystyle {{\frac {1}{2{\sqrt {x}}(1+x)}}.}\end{array}}$

Final Answer:
(a)    ${\frac {-2\pi ^{2}(\sec \theta \tan \theta -2\cos(2\theta ))}{(\sec \theta -\sin 2\theta )^{3}}}$
(b)    ${\frac {1}{2{\sqrt {x}}(1+x)}}$

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