How do you differentiate #f(x)=x^3 * sin^2x# using the product rule?

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Answer 1 · 2016-01-23T15:47:15

#f'(x)=x^2sinx(3sinx+2xcosx)#

The product rule states that

#f'(x)=sin^2xd/dx[x^3]+x^3d/dx[sin^2x]#

Now, find both of those derivatives.

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

The following will require the power rule with chain rule.

#d/dx[sin^2x]=2sinxd/dx[sinx]=2sinxcosx#

Plug these back in to the original equation.

#f'(x)=3x^2sin^2x+2x^3sinxcosx#

This can be factored, but not exceptionally helpfully:

#f'(x)=x^2sinx(3sinx+2xcosx)#