Question

How do you differentiate `f(x)= 5sinx*(1-2x)(x+1)-3` using the product rule?

Answer 1

`f'(x) =-20xsin(x) -5sin(x)-10x^2cos(x)-5xcos(x)+5cos(x)`

Explanation:

Product Rule:
`[f(x)g(x)]'=f'(x) g(x)+f(x)g'(x)`
Set `f(x)=5sin(x)` and `g(x)=(1-2x)(x+1)`

(1)
`d/dx[5sin(x)]=5cos(x)`

(2)
`d/dx[(1-2x)(x+1)]=d/dx[-2x^2-x+1]=-4x-1`

Putting it all together
`f'(x) =d/dx[5sin(x)(1-2x)(x+1)]-d/dx[3]`
`f'(x) =(5sin(x)(-4x-1))+5cos(x)(-2x^2-x+1)-0`
`f'(x) =-20xsin(x) -5sin(x)-10x^2cos(x)-5xcos(x)+5cos(x)`