Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 67

Find a.) $F'(x)$ and b.) $G'(x)$ if $f(x) = f(\cos x)$ and $G(x) = \cos (f(x))$ suppose $f$ is differentiable everywhere.


$
\begin{equation}
\begin{aligned}

\text{ a.) } F'(x) = \frac{d}{dx} [F(x)] =& f' (\cos x) \cdot \frac{d}{dx} (\cos x) \cdot \frac{d}{dx} (x)
\\
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F'(x) = \frac{d}{dx} [F(x)] =& f' (\cos x) (- \sin x) (1)
\\
\\
F'(x) = \frac{d}{dx} [F(x)] =& - \sin x f'(\cos x)
\\
\\
\\
\text{ b.) }G'(x) = \frac{d}{dx} [G(x)] =& - \sin (f(x)) \cdot \frac{d}{dx} (f(x)) \cdot \frac{d}{dx} (x)
\\
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G'(x) = \frac{d}{dx} [G(x)] =& - \sin (f(x)) \cdot f'(x) \cdot 1
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G'(x) = \frac{d}{dx} [G(x)] =& - \sin (f(x)) f'(x)

\end{aligned}
\end{equation}
$