# How do you differentiate #g(x) = x^3e^(2x)# using the product rule?

##### 1 Answer

Jan 2, 2016

#### Answer:

#### Explanation:

According to the product rule,

#d/dx[f(x)g(x)]=f'(x)g(x)+g'(x)f(x)#

Thus,

#g'(x)=e^(2x)d/dx(x^3)+x^3d/dx(e^(2x))#

Find each derivative.

#d/dx(x^3)=3x^2#

This will require the chain rule:

#d/dx(e^u)=u'e^u# , so

#d/dx(e^(2x))=e^(2x)d/dx(2x)=2e^(2x)#

Plug these back in to find

#g'(x)=3x^2e^(2x)+2x^3e^(2x)#

Optionally factored:

#g'(x)=x^2e^(2x)(2x+3)#