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

1 Answer
Feb 18, 2016

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.