# How do you find the derivative of 1/sqrt (x-1)?

Jun 18, 2016

It is $- \frac{1}{2 \sqrt{{\left(x - 1\right)}^{3}}}$.

#### Explanation:

You can use the chain rule and the derivative of the power.

Your function can be written as

$\frac{1}{\sqrt{x - 1}} = {\left(x - 1\right)}^{- \frac{1}{2}}$

we know that the derivative of ${x}^{n}$ is $n {x}^{n - 1}$.
In this case we do not have ${x}^{- \frac{1}{2}}$ but we have ${\left(x - 1\right)}^{- \frac{1}{2}}$.
So we have to apply the chain rule and write

$\frac{d}{\mathrm{dx}} {\left(x - 1\right)}^{- \frac{1}{2}} = - \frac{1}{2} {\left(x - 1\right)}^{- \frac{1}{2} - 1} \cdot \frac{d}{\mathrm{dx}} \left(x - 1\right)$

$= - \frac{1}{2} {\left(x - 1\right)}^{- \frac{3}{2}} \cdot 1$

$= - \frac{1}{2 \sqrt{{\left(x - 1\right)}^{3}}}$.