# How do you differentiate: y=(x^4)(e^x)?

May 21, 2015

Use the product rule, together with the fact that $\frac{d}{\mathrm{dx}} \left({e}^{x}\right) = {e}^{x}$

I use the product rule in the order: $\left(f g\right) ' = f ' g + f g '$

$y = {x}^{4} {e}^{x}$

$y ' = 4 {x}^{3} {e}^{x} + {x}^{4} {e}^{x}$

(Yes, I know its hard to tell where I took the derivative of ${e}^{x}$ I did it in the second term.)

Here it is again with derivatives in $\textcolor{red}{\text{red}}$

$y = {x}^{4} {e}^{x}$

$y ' = \textcolor{red}{4 {x}^{3}} {e}^{x} + {x}^{4} \textcolor{red}{{e}^{x}}$

Of course we can factor if it seems like a good idea:

$y ' = {x}^{3} {e}^{x} \left(4 + x\right)$