Question

What is the derivative of `f(x) = sin2x*cos2x`?

What is the derivative of `f(x) = sin2x*cos2x`?

Answer 1

`d/dx[f(x)]=-2[sin^2(2x)-cos^2(2x)]`

Explanation:

Some basic rules of differentiation are as follows, where `u` and `v` are functions of `x`:

1. Addition / subtraction rule
   If `y=u+-v`, `dy/dx=d/dx(u)+-d/dx(v)`

2. Chain rule
   `dy/dx=dy/(du)xx(du)/dx`

3. Product rule
   If `y=uv`, `dy/dx=u(dv)/dx+v(du)/dx`

4. Quotient rule
   If `y=u/v`, `dy/dx=(v(du)/dx-u(dv)/dx)/v^2`

Let's get started,

`f(x) = sin2x*cos2x`
`d/dx[f(x)]=(sin2x)[d/dx(cos2x)]+(cos2x)[d/dx(sin2x)]` ( Product rule )
`color(white)(d/dx[f(x)])=(sin2x)(-2)(sin2x)+(cos2x)(2)(cos2x)`
`color(white)(d/dx[f(x)])=-2[sin^2(2x)-cos^2(2x)]`