Question

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

Answer 1

By Product Rule,

`y'=(1/{2sqrt{x}}+sqrt{x})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^{1/2}cdot e^x`

By Product Rule,

`y'=1/2x^{-1/2}cdot e^x+x^{1/2}cdot e^x`

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

`=1/{2sqrt{x}}e^x+sqrt{x}e^x`

by factoring out `e^x`,

`=(1/{2sqrt{x}}+sqrt{x})e^x`