Single Variable Calculus, Chapter 3, Review Exercises, Section Review Exercises, Problem 30

Find $y'$ of $y = \sqrt{\sin\sqrt{x}}$


$
\begin{equation}
\begin{aligned}
y' &= \frac{d}{dx} \left( \sqrt{\sin\sqrt{x}} \right)\\
\\
y' &= \frac{d}{dx} (\sin \sqrt{x})^{\frac{1}{2}}\\
\\
y' &= \frac{1}{2} ( \sin \sqrt{x} )^{\frac{-1}{2}} \frac{d}{dx} ( \sin \sqrt{x})\\
\\
y' &= \frac{1}{2} ( \sin \sqrt{x} )^{\frac{-1}{2}} (\cos \sqrt{x}) \frac{d}{dx} ( \sqrt{x})\\
\\
y' &= \frac{1}{2} ( \sin \sqrt{x} )^{\frac{-1}{2}} (\cos \sqrt{x}) \frac{d}{dx} (x)^{\frac{1}{2}}\\
\\
y' &= \frac{1}{2} ( \sin \sqrt{x} )^{\frac{-1}{2}} (\cos \sqrt{x}) \left( \frac{1}{2} \right) (x)^{\frac{-1}{2}}\\
\\
y' &= \frac{\cos\sqrt{x}}{4(\sin\sqrt{x})^{\frac{1}{2}}(x)^{\frac{1}{2}}}\\
\\
y' &= \frac{\cos\sqrt{x}}{4\sqrt{x\sin\sqrt{x}}}
\end{aligned}
\end{equation}
$