Derivative of cos²x is (-2cosxsinx) which is equal to (-sin 2x). Cos²x is square of trigonometric function cos x. Derivative refers to the process of finding the change in the cos²x function with respect to the independent variable.

In this article, we will discuss the derivative of cos²x with various methods to find it including the first principle of differentiation, chain rule, and the product rule, solved examples, and some practice problems on it.

What is Derivative in Math?

Derivative of a function is the rate of change of the function to any independent variable. The derivative of a function f(x) is denoted as f′(x) or (d/dx)​[f(x)]. Derivative of trigonometric functions is easily found using various differentiations formulas.

Read More:

• Calculus in Maths
• Derivatives

What is Derivative of Cos²x?

Derivative of cos²x is -2cosxsinx. Cos²x is a composite function involving an algebraic operation on a trigonometric function. Derivative of a function gives the rate of change in the functional value for the input variable, i.e. x.

In chain rule, if we need to find the derivative of f(g(x)), it is given as f'(g(x)) × g'(x). The chain rule is one of the most fundamental and used concepts in differential calculus. Formula for the derivative of cos²x can be written as follows:

Derivative of cos²x Formula

Formula for derivative of cos²x is added below as,

d/dx[cos²x] = -2cosx.sinx

(cos²x)’ = -2cosx.sinx

We can derive it using the below-mentioned methods:

• First Principle of Differentiation
• Chain Rule
• Product Rule

Let us discuss these methods in detail one by one as follows.

Proof of Derivative of cos²x

Formula for derivative of cos²x can be derived using any of following methods:

• First principle of Differentiation
• Chain Rule
• Product Rule

Derivative of cos²x using First Principle of Derivatives

First principle of differentiation state that derivative of a function f(x) is defined as,

f'(x) = lim_h→0 [f(x + h) – f(x)]/[(x + h) – x]

f'(x) = lim_h→0 [f(x + h) – f(x)]/ h

Putting f(x) = cos²x, to find derivative of cos²x, we get,

⇒ f'(x) = limh→0 [cos²(x + h) – cos²x]/ h

⇒ f'(x) = limh→0 (cos(x+h) + cos(x)).(cosxcosh – sinxsinh – cosx)/h

Using, cos(A + B) = cosAcosB – sinAsinB

⇒ f'(x) = limh→0 (cos(x+h) + cos(x)).(cosx.(cosh – 1) – sinxsinh)/h

Now, putting limh→0(1-cosh)/h = 0 and limh→0(sinh)/h = 1

⇒ f'(x) = limh→0 (cos(x+h) + cos(x)).(-sinx)

⇒ f'(x) = (cos(x+0) + cos(x)).(-sinx)

⇒ f'(x) = (2cosx).(-sinx)

f'(x) = -2cosx.sinx

Derivative of cos²x using Chain Rule of Differentiation

Chain Rule of differentiation states that for a composite function f(g(x)),

[f{g(x)}]’ = f'{g(x)} × g'(x)

Therefore applying chain rule to f(x) = cos²x, we get,

⇒ f'(x) = 2cosx × (cosx)’

⇒ f'(x) = 2cosx × (-sinx)

f'(x) = -2cosx.sinx

Derivative of cos²x Using Product Rule

Product rule in differentiation states that,

For two functions u and v the differentiation of (u.v) is found as,

(u.v)’ = (u.v’ + u’.v)

Now f(x) = cos²x can be written as f(x) = cosx.cosx

Applying product rule for f(x) = cosx.cosx, we get,

⇒ f'(x) = (cosx.(cosx)’ + (cosx)’.sinx)

⇒ f'(x) = (cosx.(-sinx) + (-sinx).cosx)

f'(x) = -2cosx.sinx

Derivative of cos²x using Chain Rule of Differentiation

Chain Rule of differentiation states that for a composite function f(g(x)),

[f{g(x)}]’ = f'{g(x)} × g'(x)

Therefore applying chain rule to f(x) = cos²x

⇒ f'(x) = 2cosx × (cosx)’

⇒ f'(x) = 2cosx × (-sinx)

⇒ f'(x) = -2cosx.sinx

Thus, we have derived the derivative of f(x) = cos²x using the chain rule.

Also, Check

• Derivative of Sec²x
• Derivative of Inverse Trigonometric Functions

Examples on Derivative of cos²x

Some examples related to derivative of cos²x are,

Example 1: Find the derivative of f(x) = cos²(x²+4)

Solution:

We have, f(x) = cos²(x²+4)

By applying chain rule,

⇒ f'(x) = -2cos(x²+4)×sin(x²+4)×(x²+4)’

⇒ f'(x) = -2cos(x²+4)×sin(x²+4)×(2x)

⇒ f'(x) = -4x.cos(x²+4).sin(x²+4)

Example 2: Find the derivative of f(x) = sec²x

Solution:

Here, f(x) = sec²x can be written as, f(x) = 1/cos²x,

By applying quotient rule, we get,

⇒ f'(x) = (cos²x(1)’ – (1)(cos²x)’)/(cos⁴x)

⇒ f'(x) = [-2cosx.(-sinx)]/(cos⁴x)

On simplification, we get

⇒ f'(x) = 2sec²x.tanx

Example 3: Find the derivative of f(x) = xcos²x

Solution:

For f(x) = xcos²x, by applying product rule, we get,

⇒ f'(x) = x(cos²x)’ + (x)’cos²x

⇒ f'(x) = x.(-2cosx.sinx) + cos²x

⇒ f'(x) = cosx.(-2xsinx + cosx)

Practice Problems on Derivative of cos²x

Various practice questions related to derivative of e^2x are,

Q1: Find the derivative of the function f(x) = cos²(x²+4x)

Q2: Find the derivative of the function f(x) = sec²x + cos²x

Q3: Find the value of f'(x), if f(x) = cos⁴x

Q4: If y = cos²x – sin²x, then find the value of dy/dx.

Q5: If y = (cos²x)/x, find the value of dy/dx.

Summary

To find the derivative of [Tex]\cos^2(x)[/Tex], we use the chain rule. Let [Tex]u = \cos(x)[/Tex], so that cos²(x)) becomes u². The derivative of u² with respect to u is 2u, and the derivative of cos⁡(x) with respect to x is −sin⁡(x). Applying the chain rule, we multiply these derivatives: [Tex]\frac{d}{dx} (\cos^2(x)) = 2\cos(x) \cdot (-\sin(x))[/Tex]. Simplifying this, we get [Tex]\frac{d}{dx} (\cos^2(x)) = -2\cos(x)\sin(x),[/Tex] which can also be written as [Tex]-\sin(2x)[/Tex] using the double-angle identity [Tex]\sin(2x) = 2\sin(x)\cos(x)[/Tex]. Therefore, the derivative of [Tex]\cos^2(x) [/Tex]is [Tex]-\sin(2x)[/Tex].

FAQs on Derivative of cos²x

What is derivative?

Derivative of a function is defined as the rate of change of the function with respect to a variable.

What is formula for derivative of cos²x.

Formula for the derivative of cos²x is: (d/dx) cos²x = -2cosx.sinx

What are different methods to prove derivative of cos²x?

Different methods to prove derivative of cos²x are:

• By using First Principle of Derivative
• By Product Rule
• By Chain Rule

What is formula for cos square x?

Formula for cos²x in trigonometry is cos²x = 1 – sin²x.

What is derivative of cos square x cube?

Derivative of cos square x cube is, d(cos²(x³))/dx = -3 sin(2x³).

What is derivative of sin square x?

The derivative of sin square x is 2 sin 2x