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

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Answer 1 · 2016-01-02T22:34:45

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

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)#.

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

Optionally factored:

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