Chain rule for the derivative of a composite function

Question

$$y=(\sin x)^{\sqrt{x}}.$$

I know that I suppose to apply the chain rule here, but I can't get clearly what
is the composition here.

Answer 1

I would probably use logarithmic differentiation here: \begin{align*} y&=(\sin(x))^{\sqrt{x}} \\ \ln(y)&=\sqrt{x}\,\ln(\sin(x)) \\ \frac1y\,\frac{dy}{dx}&=\frac12\,x^{-1/2}\,\ln(\sin(x))+\sqrt{x}\,\frac{\cos(x)}{\sin(x)}. \end{align*} Can you finish?

The problem with the straight-forward approach is that you don't have a power rule, nor do you have an exponent rule you can use. It's a bit analogous to differentiating $x^x$.

Answer 2

It's easier if you write $$y=e^{\sqrt{x}\ln(\sin(x))}$$ Then $$y'=e^{\sqrt{x}\ln(\sin(x))}(\sqrt{x}\ln(\sin(x)))'=(\sin(x))^{\sqrt{x}}(\sqrt{x}\ln(\sin(x)))'$$

Now you only have to differentiate $\sqrt{x}\ln(\sin(x))$, in which case you need the product rule and the chain rule.