How do you differentiate f (x) = cos²(2x)? 

Solution:

Differentiation is the process of finding the instantaneous rate of change of a function.The function cos² x can be expressed as cos(x) × cos(x).

To differentiate the function we will take y = cos²x. Let's see how.

Let, u = cosx   

⇒ du/dx = - sinx ------------------- (1)

Now, y = u²         (Since, y = cos²x )

⇒ dy/du = 2u --------------- (2)

By applying chain rule,

⇒ dy/dx = dy/du × du/dx 

= - sinx × 2cosx

= -2sinx.cosx

= - sin(2x)       (Using double angle identity : sin2x = 2sinxcosx )

Thus, the derivative of f(x) = cos²x is -sin2x.

How do you differentiate f (x) = cos²(2x)? 

Summary:

Differentiation of f(x) = cos²x is - sin2x