Question

How do you find the derivative of `cos^2(3x)`?

Answer 1

`d/(dx)cos^2(3x)=-6sin(3x)cos(3x)`

Explanation:

Using the chain rule, we can treat `cos(3x)` as a variable and differentiate `cos^2(3x)` in relation to `cos(3x)`.

Chain rule: `(dy)/(dx)=(dy)/(du)*(du)/(dx)`

Let `u=cos(3x)`, then `(du)/(dx)=-3sin(3x)`

`(dy)/(du)=d/(du)u^2->`since `cos^2(3x)=(cos(3x))^2=u^2`

`=2u=2cos(3x)`

`(dy)/(dx)=2cos(3x)*-3sin(3x)=-6sin(3x)cos(3x)`