Taking the derivative of a compound fractions So I need to do a bunch of stuff with the derivative of $$f(x) = \frac{1}{\sqrt{x}} .$$ Lack of sleep is murdering my ability to do compound fractions today. Can someone show me how to either rationalize this or take its derivative?

We know that differentiation of $x^n$ ($n \in \mathbb{R}$) with respect to $x$ is $n \times x^{n-1}$.

Notice that we could write: $$\frac{1}{\sqrt{x}} = \frac{1}{x^{1/2}} = x^{-1/2}$$ now applying differentiation:

$$ \frac{\mathrm{d} }{\mathrm{d} x} \bigl(x^{-1/2}\bigr)= -\frac{1}{2} \times x^{-1/2-1} = -\frac{1}{2} \times x^{-3/2}=-\frac{1}{2 \sqrt{x^{\large3}}}$$.