Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 4

Express the composite function $y = \tan(\sin x)$ 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 $u = g(x) = \sin x$ and $ y = f(u) = \tan u$

Then,


$
\begin{equation}
\begin{aligned}
y' &= \frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}\\
\\
y' &= \frac{d}{du} (\tan x) \cdot \frac{d}{dx} ( \sin x )\\
\\
y' &= ( \sec^2 u)(\cos x) && \text{Substitute value of } u \text{ and simplify.}\\
\\
y' &= \cos x \sec^2(\sin x)
\end{aligned}
\end{equation}
$

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