If y = cos2 x2, find dy/dx

Concept

To differentiate a composite function, use the chain rule: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x). Here y = cos2(x2) is a triple composition — an outer square, a middle cosine, and an inner x2 — so the chain rule is applied across all three layers, and recall d/du(cos u) = -sin u and d/dx(x2) = 2x.

Application

1. Write y = [cos(x2)]2, treating it as (something)2 with the ‘something’ = cos(x2).

2. Differentiate the outer square: dy/dx = 2·cos(x2)·d/dx[cos(x2)].

3. Differentiate the middle cosine: d/dx[cos(x2)] = -sin(x2)·d/dx[x2].

4. Differentiate the inner power: d/dx[x2] = 2x.

5. Multiply the three factors: dy/dx = 2·cos(x2)·(-sin(x2))·2x = -4x·cos(x2)·sin(x2).

Cross-check

Using the identity 2·sinθ·cosθ = sin(2θ), the result can be rewritten as -2x·sin(2x2). Differentiating y = cos2(x2) = ½(1 + cos(2x2)) instead gives dy/dx = ½·(-sin(2x2))·4x = -2x·sin(2x2), the same value — confirming dy/dx = -4x·cos(x2)·sin(x2).