009A Sample Final 3, Problem 2

Find the derivative of the following functions:

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

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

Foundations:
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:

Step 2:

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

Final Answer:
(a)
(b)
(c)

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