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