# How do you use the product rule to find the derivative of y=sqrt(x)*e^x ?

Sep 22, 2014

By Product Rule,

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

Let us look at some details.

By rewriting the square-root as the 1/2-power,

$y = \sqrt{x} \cdot {e}^{x} = {x}^{\frac{1}{2}} \cdot {e}^{x}$

By Product Rule,

$y ' = \frac{1}{2} {x}^{- \frac{1}{2}} \cdot {e}^{x} + {x}^{\frac{1}{2}} \cdot {e}^{x}$

by rewriting the 1/2-power as the square-root,

$= \frac{1}{2 \sqrt{x}} {e}^{x} + \sqrt{x} {e}^{x}$

by factoring out ${e}^{x}$,

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