# How do you differentiate f(x)=1/(sqrt(x-3) using the chain rule?

Jan 1, 2016

Chain rule states that $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$.
We'll also need two power rules here.

#### Explanation:

• ${a}^{-} n = \frac{1}{a} ^ n$

• ${a}^{\frac{m}{n}} = \sqrt[n]{{a}^{m}}$

So, we can rewrite it as $f \left(x\right) = {\left(x - 3\right)}^{- \frac{1}{2}}$

Renaming $u = x - 3$, we get $f \left(x\right) = {u}^{- \frac{1}{2}}$ and can now differentiate it.

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{2} {u}^{- \frac{3}{2}} \left(1\right) = - \frac{1}{2 \left({u}^{\frac{3}{2}}\right)} = - \frac{1}{2 \sqrt{{\left(x - 3\right)}^{3}}}$