Question

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

Answer 1

We could calculate the derivative using the product rule

`(d(f*g))/(dx) = (df)/(dx)* g+f*(dg)/(dx)`

However we can also note that:

`sin(2x)=2sinxcosx`

so:

`f(x) = cos^2(x)sin^2(x) = ((sin2x)/2)^2=1/4sin^2(2x)`

Using the chain rule:

`(d(1/4sin^2(2x)))/dx= 1/4*2sin(2x)cos(x)* 2 = 2sinx * cos x * cos x= 2sinxcos^2x`