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

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62
Dec 26, 2016

In this problem we have to use the chain rule.

$y = {e}^{\sqrt{x}} = {e}^{{x}^{\frac{1}{2}}}$, convert the square root to its rational power

Apply the chain rule and begin to simplify

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

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

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

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

Convert the exponents to positive numbers

$y ' = {e}^{\sqrt{x}} / \left(2 {x}^{\frac{1}{2}}\right)$

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

Rationalize

$y ' = {e}^{\sqrt{x}} / \left(2 \sqrt{x}\right) \cdot \left(\frac{\sqrt{x}}{\sqrt{x}}\right)$

$y ' = \frac{{e}^{\sqrt{x}} \sqrt{x}}{2 x}$

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