Question

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

Answer 1

`d/dxcos^2(x^2-2) = 2x (sin(4-2x^2))`

Explanation:

First of all define: `f(x) = cos(x^2-2)`
First use the chain rule to find the derivative of f(x):
`f'(x)= -sin(x^2-2) (2x)`

Next split `cos^2(x^2-2)` into `cos(x^2-2) cos(x^2-2)`.
In other words: `f(x)^2 = f(x)f(x)`

Now the product rule can be used to find the derivative of f(x)^2:
`f'(x)^2 = f'(x)f(x) + f(x)f'(x)`
`f'(x)^2 = (-sin(x^2-2) (2x))(cos(x^2-2)) + (cos(x^2-2))(-sin(x^2-2) (2x))`

`f'(x)^2 = 2 (cos(x^2-2))(-sin(x^2-2) (2x))`
`f'(x)^2 = -4x (cos(x^2-2)sin(x^2-2))`

Using the trigonometric identity: `cos(x)sin(x) = 1/2 sin(2x)`

We finally gain:
`f'(x)^2 = -4x (1/2(sin(2(x^2-2))`
`f'(x)^2 = -2x (sin(2x^2-4))`

SIn(x) is an odd function, so it's true that:
sin(x) = -sin(-x)
Which means we can also write the answer as:
`f'(x)^2 = 2x (sin(4-2x^2))`
Which removes the minus sign from the front, making the expression tidier.