# What is the derivative of sqrt(x - 2)?

Jun 24, 2016

${f}^{'} = \frac{\sqrt{x - 2}}{2 x - 4}$

#### Explanation:

I prefer to use a particular notation.

Let $u = x - 2 \text{ }$ Then $\text{ } \frac{\mathrm{du}}{\mathrm{dx}} = 1$

Let $\text{ "y=sqrt(x-2)" "=" "sqrt(u)" "=" } {u}^{\frac{1}{2}}$

$\frac{\mathrm{dy}}{\mathrm{du}} = \frac{1}{2} {u}^{\frac{1}{2} - 1} \to \frac{1}{2} {u}^{- \frac{1}{2}}$

But $\frac{\mathrm{dy}}{\mathrm{dx}} \text{ "=" } \frac{\mathrm{du}}{\mathrm{dx}} \times \frac{\mathrm{dy}}{\mathrm{du}}$

$\implies \frac{\mathrm{dy}}{\mathrm{du}} = 1 \times \frac{1}{2 \sqrt{x - 2}}$

Multiply by 1 in the form of $1 = \frac{\sqrt{x - 2}}{\sqrt{x - 2}}$

$\implies \frac{\mathrm{dy}}{\mathrm{du}} = \frac{\sqrt{x - 2}}{2 \left(x - 2\right)} \text{ "=" } \frac{\sqrt{x - 2}}{2 x - 4}$

Or if you prefer:

${f}^{'} = \frac{\sqrt{x - 2}}{2 x - 4}$