Question

find the derivative of cos^2 (2x)

Answer 1

To find the derivative of cos^2 (2x), we can use the chain rule.

Let u = cos(2x)
Taking the derivative of u with respect to x:
du/dx = -sin(2x)*2

Now, we want to find the derivative of cos^2 (2x) with respect to x. Using the chain rule, we have:

d/dx (cos^2 (2x)) = d/du (cos^2 (u)) * du/dx

The derivative of cos^2(u) with respect to u is:

d/du (cos^2 (u)) = 2*cos(u)*(-sin(u))

Substituting the factor for u back in, we have:

d/dx (cos^2 (2x)) = 2*cos(u)*(-sin(u)) * du/dx

Now, we substitute back in u = cos(2x):

d/dx (cos^2 (2x)) = 2*cos(cos(2x))*(-sin(cos(2x))) * (-sin(2x)*2)

This simplifies to:

d/dx (cos^2 (2x)) = 4*cos(cos(2x))*sin(cos(2x))*sin(2x)

Answer 2

what convoluted rubbish!

d/dx cos^2(2x) = 2 cos(2x) * -sin(2x) * 2
= -4 cos(2x) sin(2x)
= -2 sin(4x)

Note that this also can be achieved by noting that
cos^2(2x) = (1 + cos(4x))/2
so the derivative is
1/2 * -sin(4x) * 4
= -2 sin(4x)

Answer 3

Apologies for any confusion caused. Your method is correct. The derivative of cos^2(2x) is indeed -2sin(4x). Thank you for pointing out the mistake in the previous response.