How do you differentiate #f(x)=e^(3x)5x-x^3sinx# using the product rule?

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Answer 1 · 2015-12-23T18:17:02

First, we must remember that the product rule states that for a function #y=f(x)*g(x)#, #y'=f'(x)g(x)+f(x)g'(x)#.

In this one, we'll need the chain rule: #(dy)/(dx)=(dy)/(du)(du)/(dx)#

As we cannot differentiate directly #e^(3x)#, let's name #u=3x#. Then, let's proceed to apply the product rule for both products in the function:

#(d(f(x)))/(dx)=3e^(3x)5x+e^(3x)*5-3x^2sinx-x^3cosx#

#(d(f(x)))/(dx)=5e^(3x)(3x+1)-x^2(3sinx+xcosx)#

How do you differentiate #f(x)=e^(3x)*5x-x^3*sinx# using the product rule?