#### To determine

**To find:** The derivative of f(θ)=cos(θ2).

#### Answer

The derivative of f(θ)=cos(θ2) is f′(θ)=−2θsinθ2.

#### Explanation

**Given:**

The function is f(θ)=cos(θ2).

**Result 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:**

Obtain the derivative of f(θ).

Let u(θ)=θ2 and g(u)=cosu.

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

f′(θ)=(g∘u)′(θ)=g′(u(θ))⋅u′(θ)=ddu(g(u))⋅ddθ(u(θ))=ddu(cosu)⋅ddθ(θ2) [Qg(u)=cosu u(θ)=θ2]

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

f′(θ)=(−sinu)(2θ2−1)=−sinu(2θ)=−2θsinu=−2θsinθ2 [Qu=θ2]

Therefore, the derivative of f(θ)=cos(θ2) is f′(θ)=−2θsinθ2.