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

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Answer 1 · 2014-09-22T08:22:20

#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#