Question

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?

Answer 1

`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))`