Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 6

Express the composite function $y = \sin\sqrt{x}$ 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 $u = g(x) = \sqrt{x}$ 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}{dx} (\sqrt{x})\\
\\
y' &= (\cos u) \left( \frac{1}{2\sqrt{x}}\right) && \text{Simplify the equation}\\
\\
y' &= \frac{\cos u}{2\sqrt{x}} && \text{Substitute value of } u \\
\\
y' &= \frac{\cos \sqrt{x}}{2\sqrt{x}}

\end{aligned}
\end{equation}
$