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

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Answer 1 · 2015-12-01T21:35:48

#f'(x)=x^2e^x(3sin(x)+xsin(x)+xcos(x))#

According to the product rule:

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

#f'(x)=e^xsin(x)d/dx[x^3]+x^3sin(x)d/dx[e^x]+x^3e^xd/dx[sin(x)]#

Find each derivative separately.

#d/dx[x^3]=3x^2#

#d/dx[e^x]=e^x#

#d/dx[sin(x)]=cos(x)#

Plug back in.

#f'(x)=3x^2e^xsin(x)+x^3e^xsin(x)+x^3e^xcos(x)#

#f'(x)=x^2e^x(3sin(x)+xsin(x)+xcos(x))#