How do you differentiate #g(x) = x^3e^(2x)# using the product rule?
1 Answer
Jan 2, 2016
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)#