# How do you find the derivative of  y=e^(x^(1/2))?

##### 1 Answer
Mar 9, 2018

${e}^{\sqrt{x}} / \left(2 \sqrt{x}\right)$

#### Explanation:

A substitution here would help tremendously!

Let's say that ${x}^{\frac{1}{2}} = u$

now,

$y = {e}^{u}$

We know that the derivative of ${e}^{x}$ is ${e}^{x}$ so;

$\frac{\mathrm{dy}}{\mathrm{dx}} = {e}^{u} \cdot \frac{\mathrm{du}}{\mathrm{dx}}$ using the chain rule

$\frac{d}{\mathrm{dx}} {x}^{\frac{1}{2}} = \frac{\mathrm{du}}{\mathrm{dx}} = \frac{1}{2} \cdot {x}^{- \frac{1}{2}} = \frac{1}{2 \sqrt{x}}$

Now plug $\frac{\mathrm{du}}{\mathrm{dx}}$ and $u$ back into the equation :D

$\frac{\mathrm{dy}}{\mathrm{dx}} = {e}^{\sqrt{x}} / \left(2 \sqrt{x}\right)$