Question

What is the derivative of `sin^2(x) cos (x)`?

Answer 1

`dy/dx = sinx(3cos^2x- 1)`

Explanation:

`y = (1 - cos^2x)cosx = cosx - cos^3x`

We know the derivative of `cosx` is `-sinx`. Letting `y = u^3` and `u = cosx`, we have: `(cos^3x)' = -sinx3u^2 = -sinx3(cosx)^2 =-3cos^2xsinx`

The derivative of the entire expression is:

`dy/dx = -sinx - ( -3cos^2xsinx)`

`dy/dx = 3cos^2xsinx - sinx`

`dy/dx= sinx(3cos^2x- 1)`

Hopefully this helps!

Answer 2

`sin x(3 cos^2 x -1)`

Explanation:

`d/dx (sin^2 x cos x)= sin^2 x d/dx cosx + cos x d/dx sin^2 x`

= `-sin^2 x sinx +cos x (2sinx d/dx sinx)`

= `- sin^3 x +2sin x cos^2 x`

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

=`sin x (cos^2x -1 + 2 cos^2 x)`

= `sin x(3 cos^2 x -1)`