Derivative of Cos Square x
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.
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) = limh→0 [f(x + h) – f(x)]/[(x + h) – x]
f'(x) = limh→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
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 e2x 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.
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