# What is the derivative of y=3x^2e^(5x) ?

Jul 24, 2014

This is a product of the function $3 {x}^{2}$ and the function ${e}^{5 x}$, which is itself the composite of the functions given by ${e}^{x}$ and by $5 x$.

Thus we will need the product rule to the effect that:

$\left(3 {x}^{2} {e}^{5 x}\right) ' = \left(3 {x}^{2}\right) ' {e}^{5 x} + 3 {x}^{2} \left({e}^{5 x}\right) '$

As well as the fact that:

$\left(3 {x}^{2}\right) ' = 6 x$

Using these basic differentiation rules, and the Chain Rule combined with $\left({e}^{x}\right) ' = {e}^{x}$, we can calculate that:

$\left({e}^{5 x}\right) ' = {e}^{5 x} \setminus \times \left(5 x\right) ' = 5 {e}^{5 x}$.

As a result:

$\left(3 {x}^{2} {e}^{5 x}\right) ' = 6 x {e}^{5 x} + 15 {x}^{2} {e}^{5 x}$.