Y = $\displaystyle \displaystyle \sqrt{(x^3+5)}$
$\displaystyle \dfrac{d}{dx}\left(\sqrt{u(x)}\right)=\dfrac{1}{2\ sqrt{u(x)}}\cdot u'(x)$
Fernando Revilla
Edited: Sorry, I didn't see e^(i*pi)'s post.
Let's see!
$\displaystyle \displaystyle\ u=x^3+5\Rightarrow\frac{du}{dx}=3x^2$
From the Chain Rule
$\displaystyle \displaystyle\frac{dy}{dx}=\frac{dy}{du}\;\frac{du }{dx}=\frac{d}{du}u^{\frac{1}{2}}\;3x^2=\frac{1}{2 }u^{-\frac{1}{2}}\;3x^2=\frac{1}{2u^{\frac{1}{2}}}3x^2= \frac{3x^2}{2\sqrt{x^3+5}}$
So you see, those fast methods can be quite useful as an alternative to messing with fractional indices!