Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 1

Express the composite function $y = \sin 4x$ in the form $f(g(x))$. [Identify the inner function $u=g(x)$ and the outer function $y = f(u)$.] Then find the derivative $\displaystyle \frac{dy}{dx}$


Let $y = f(g(x))$ where $u=g(x)=4x$ and $y = f(u) = \sin u$

Then

$
\begin{equation}
\begin{aligned}
y' &= \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}\\
\\
y' &= \frac{d}{du}(\sin u) \cdot \frac{d}{du} (4x)\\
\\
y' &= (\cos u) (4) && \text{Simplify the equation}\\
\\
y' &= 4 \cos u && \text{Substitute value of } u\\
\\
y' &= 4 \cos 4x
\end{aligned}
\end{equation}
$