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

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.