will someone do a step by step so I can try and teach myself this stuff..
Let
$\displaystyle \frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2}$
The chain rule:
$\displaystyle \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
Let $\displaystyle f(x) = \tan^{-1} x, g(x) = \cos (7x)$
$\displaystyle \frac{d}{dx} \tan^{-1} (\cos (7x)) =$
$\displaystyle \frac{1}{1 + (\cos (7x))^2} \cdot \frac{d}{dx} \cos (7x) = $
$\displaystyle \frac{-\sin (7x)}{1 + (\cos (7x))^2} \cdot \frac{d}{dx} (7x) =$
$\displaystyle \frac {-7 \sin (7x)}{1 + (\cos (7x))^2}$