Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 44

Determine the derivative of the function $y = \cos^4(\sin^3 x)$


$
\begin{equation}
\begin{aligned}
y' &= \frac{d}{dx} \left[ \cos^4(\sin^3x)\right]\\
\\
y' &= \frac{d}{dx} \left[ \cos(\sin^3x)\right]^4\\
\\
y' &= 4\left[ \cos(\sin^3x)\right]^3 \frac{d}{dx} \left[ \cos(\sin^3x)\right]\\
\\
y' &= 4 \cos^3(\sin^3x) \left[ -\sin ( \sin^3x)\frac{d}{dx}(\sin^3x)\right]\\
\\
y' &= 4 \cos^3(\sin^3x) \left[ -\sin(\sin^3x)\frac{d}{dx}(\sin x)^3\right]\\
\\
y' &= 4 \cos^3(\sin^3x) \left[ -\sin(\sin^3x) \cdot 3(\sin x)^2 \frac{d}{dx} (\sin x) \right]\\
\\
y' &= 4 \cos^3(\sin^3x) \left[ -\sin (\sin^3x) \cdot 3 \sin^2x \cos x\right]\\
\\
y' &= 4 \cos^3(\sin^3x) \left[ -3\sin (\sin^3x) \sin^2x\cos x\right]\\
\\
y' &= -12\cos^3(\sin^3x)\sin(\sin^3x)\sin^2x\cos x
\end{aligned}
\end{equation}
$