Answer
$y'=\dfrac{5}{2}z^{3/2}+\dfrac{1}{2}z^{-1/2}e^{z}+z^{1/2}e^{z}$
Work Step by Step
$y=(z^{2}+e^{z})\sqrt{z}$
Rewrite the function like this:
$y=(z^{2}+e^{z})z^{1/2}$
Differentiate using the product rule:
$y'=(z^{2}+e^{z})(z^{1/2})'+(z^{1/2})(z^{2}+e^{z})'=...$
$...=(z^{2}+e^{z})(\dfrac{1}{2}z^{-1/2})+(z^{1/2})(2z+e^{z})=...$
Evaluate the products and simplify
$...=\dfrac{1}{2}z^{3/2}+\dfrac{1}{2}z^{-1/2}e^{z}+2z^{3/2}+z^{1/2}e^{z}=...$
$...=\dfrac{5}{2}z^{3/2}+\dfrac{1}{2}z^{-1/2}e^{z}+z^{1/2}e^{z}$
