If f(x)=5x(sinx)(cosx), then what does f'(x) equal?

No idea how to take the derivative of this. I thought it would be 5(cosx)(-sinx), but my online homework program tells me this is wrong. How do I find the derivative here?

1 Answer
Mar 6, 2018

f'(x)=5(cos(x)sin(x)+x(cos^2(x)-sin^2(x))

Explanation:

We want to find the derivative of

f(x)=5xsin(x)cos(x)

With repeated use of the product, if f=g*h*q then

f'=(g*h*q)'=(g*h)'*q+g*h*q'

=(g'*h+g*h')*q+g*h*q'

=g'*h*q+g*h'*q+g*h*q'

Here

  • g=5x=>g'=5

  • h=cos(x)=>h'=-sin(x)

  • q=sin(x)=>q'=cos(x)

f'(x)=5cos(x)sin(x)-5xsin(x)sin(x)+5xcos(x)cos(x)

=5cos(x)sin(x)-5xsin^2(x)+5xcos^2(x)

=5(cos(x)sin(x)+x(cos^2(x)-sin^2(x))