#### To determine

**To find:** The derivative of g(θ)=cos2θ.

#### Answer

The derivative of g(θ)=cos2θ is −2cosθsinθ_

#### Explanation

**Given:**

The function is g(θ)=cos2θ.

**Formula used:**

**The Chain Rule:**

If *g* is differentiable at θ and *f* is differentiable at g(θ), then the composite function F=f∘g defined by F(θ)=f(g(θ)) is differentiable at θ and F′ is given by the product

F′(θ)=f′(g(θ))⋅g′(θ) (1)

**Power Rule:**

If *n* is positive integer, then ddx(xn)=nxn−1 (2)

**Calculation:**

Let h(θ)=cosθ and f(u)=u2 where u=h(θ).

Apply the chain rule as shown in equation (1),

g′(θ)=f′(h(θ))⋅h′(θ) (3)

The derivative f′(h(θ)) is computed as follows,

f′(h(θ))=f′(u) [Qu=h(θ)]=ddu(f(u))=ddu(u2)

Apply the power rule as shown in equation (2),

f′(h(θ))=(2u2−1)=2u=2cosθ

The derivative is f′(h(θ))=2cosθ.

The derivative of h(θ) is computed as follows,

h′(θ)=ddθ(h(θ))=ddθ(cosθ)=−sinθ

The derivative of h(θ) is h′(θ)=−sinθ.

Substitute 2cosθ for f′(h(θ)) and −sinθ for h′(θ) in equation (3),

g′(θ)=2cosθ(−sinθ)=−2cosθsinθ

Therefore, the derivative of g(θ)=cos2θ is g′(θ)=−2cosθsinθ_.