how to differentiate sin^3x ??

Question

the answer is 3sin^2 (2x) but i dont know how to get it

Answer 1

Let y=sin(x), so dy/dx=cos(x); and let f(x)=sin³(x)=y³.

df/dx=(df/dy)(dy/dx)=3y²cos(x)=3sin²(x)cos(x).

sin(2x)=2sin(x)cos(x); sin(x)sin(2x)=2sin²(x)cos(x); sin²(x)cos(x)=½sin(x)sin(2x).

The derivative can be written df/dx=(3/2)sin(x)sin(2x).

sin²(2x)=(sin(2x))²=4sin²(x)cos²(x) is not the derivative.