Derivative of Sin 2x

Derivative of sin 2x is 2.cos 2x. Sin 2x is a trigonometric function in which the angle of sin is represented as twice an angle. The trigonometric expansion of sin 2x is 2sinxcosx. The derivative of sin 2x is the rate of change in the function sin 2x to the independent variable x.

In this article, we will learn what is derivative of sin 2x is and how to differentiate sin 2x using various methods in calculus.

What is the Derivative of Sin 2x

Derivative of sin 2x refers to the rate of change of sin 2x to the independent variable x. Sin 2x formula is one of the double-angle formulas in trigonometry. Using this formula, we can find the sine of the angle whose value is doubled. The most commonly used formula of sin 2x is twice the product of the sine function and cosine function which is mathematically given by, sin 2x = 2 sin x cos x.

Derivative of Sin 2x Formula

Derivative of sin 2x is given as

d/dx(sin 2x) = 2.cos 2x

Learn,

• Derivatives in Maths
• Trigonometric Functions

Sin 2x Formula Proof

The formula for sin 2x can be derived by using the angle sum formula for sine.

Let us see the derivation of sin 2x step by step:

We know that the sum formula of sin is sin(A + B) = sin A cos B + sin B cos A.

Substitute A = B = x in the formula sin(A + B) = sin A cos B + sin B cos A, we get

sin(x + x) = sin x cos x + sin x cos x

⇒ sin 2x = 2 sin x cos x

Hence, we have derived the formula of sin 2x.

Sin 2x derivative in terms of Tan

We can also write the formula of sin 2x in terms of tan or tangent function only.

We already know the sin 2x formula, sin 2x = 2 sin x cos x

Multiplying and dividing the above equation by cos x, we get

sin 2x = (2 sin x cos2x)/(cos x)

= 2 (sin x/cosx ) × (cos2x)

We know that sin x/cos x = tan x and cos x = 1/(sec x). So

sin 2x = 2 tan x × (1/sec2x)

Using Pythagorean trigonometric identities, sec2x = 1 + tan2x. Substituting this, we have

sin 2x = (2tan x)​/(1 + tan2x)

Therefore, the sin 2x formula in terms of tan is sin 2x = (2tan x)​/(1 + tan2x).

How to Find Derivative of Sin 2x

Here we are going to find the derivative of Sin 2x proof by three methods – First Principle, Chain Rule, Product Rule.

Derivative of Sin 2x Proof by First Principle

To differentiate Sin 2x by First Principal , lets assume f(x)=sin 2x.

Thus consequently using the first principle, we get

⇒ f(x) = lim _h->0 [f(x+h)-f(x)]/h

Thus, we can find the derivative as,

⇒f(x) = lim _h->0 [sin 2(x+h) – sin 2(x)]/h

⇒f(x) = lim _h->0 [sin 2x cos 2h + cos 2x sin 2h – sin 2x]/h , since [sin(A+B) = sin A cos B + cos A sin B]

⇒f(x) = lim _h->0 [- sin 2x(1 – cos 2h) + cos 2x sin 2h]/h

Using the half angle formula, 1- cos 2h = 2 sin2 (h),

⇒f(x) = (-sin 2x) { lim [(2 sin2 (h))]/h} + (cos 2x) {lim (sin 2h)/2h}

⇒f(x) = (-sin 2x) [lim (sin(h))/(h). lim sin (h)] + (cos 2x) {lim (sin 2h)/2h}

⇒f(x) = 0 + cos2x (2), since [limx->0 sin 2x/2x = 2]

⇒f(x) = 2 cos 2x

Thus, the required derivative of sin 2x is 2 cos 2x.

Derivative of Sin 2x Proof by Chain Rule

The chain rule computes the derivative of a function combination of two or more functions.

We know the chain rule is,

d/dx(f(g(x))) = d/d(g(x)) [f(g(x)). d/dx (g(x))]

Applying the derivative we get,

⇒ d/dx(sin(2x))

⇒ d/d(2x)(sin(2x)).d/dx(2x)

⇒ cos(2x).2

⇒ 2cos(2x)

Thus using chain rule sin 2x = 2 cox 2x.

Derivative of Sin 2x Proof by Product Rule

Using product rule we easily find the derivative of sin 2x as,

Using the sin double angle formula, Sin2x = 2 sinx cosx

Let u = 2 sin x, and v = cos x. Thus,

⇒ u’ = 2 cos x

⇒ v’ = – sin x

According to product rule:

f(x) = uv’ + vu’

⇒ (2sinx)(–sinx) + (cosx)(2cosx)

⇒ 2(cos2x–sin2x)

⇒ 2cos2x

Thus, using product rule proved that sin 2x = 2 cos 2x.

nth Derivative of Sin 2x

The nth derivative of sin 2x is obtained by differentiating sin 2x n times. To determine nth derivative, we first need first derivative, then second derivative and so on.

Leibniz Formula expresses the derivative of the product of two functions of the nth order. Assuming that the derivatives of the functions u (x) and v (x) are up to nth order

⇒ (uv)⁽ⁿ⁾=∑ⁿ_i=0 (ⁿ_i)u⁽ⁿ⁻ⁱ⁾v⁽ⁱ⁾

Thus, the nth derivative of sin 2x is:

⇒ (2sinxcosx)⁽ⁿ⁾ = 2 ∑ⁿ_i=0 (ⁿ_i) [sinx]ⁿ⁻ⁱ[cosx]ⁱ

Here, (ⁿ_i) indicates the number of i -combinations of elements.

Learn,

• Differentiation of Trigonometric Functions
• Differentiation Formulas
• Trigonometry Formulas

Examples on Derivative of sin 2x

Example 1 : If cos A = 3/5 where A is in quadrant I, then find the value of sin 2A?

Solution:

According to Pythagorean identity

sin2A + cos2A = 1

sin2A = 1 – cos2A

sin A = ±√(1 − cos2A)

sin A = ±√(1 − (3/5)2)

sin A = ±√(16/25)

sin A = ± 4/5

Since A is in quadrant I, sin⁡ A is positive. Thus,

sin A = 4/5

From sin 2x formula, sin 2x = 2 sin x cos x. From this,

sin 2A = 2 sin A cos A

= 2 (4/5) (3/5)

= 24/25

Example 2: Find the value of sin 90 Degrees using the Sin2x Formula.

Solution:

Sin2x Formula is given as Sin (2x) = 2Sin x Cos x. To find the value of sin 90 degrees, Sin2x Formula can to be used.

2x = 90^o

x = 90°/2

x = 45°

Value of x is obtained. Substituting its value into the Sin2x Formula,

Sin (2 x 45°) = 2sin45° cos45°

We know that sin45° = 1/√2 and cos 45° = 1/√2. Using these values we get,

Sin 90°=2×1/√2 x 1/√2

Sin 90° = 1

Example 3: Determine the value of sin 2x if sin x = 4/5

Solution:

Given that, sin x = 4/5, Using the Pythagorean theorem we can obtain that, cos x = 3/5.

According to Sin2x formula we get,

sin 2x = 2 sin x cos x

Putting the given sin x value and cos x value, we get

sin 2x = 2 (4/5) (3/5)

sin 2x = 24/25

Example 4: Calculate the derivative of sin(2x+1)?

Solution:

Let f(x) = sin(2x+1)

Applying Chain rule,

⇒f'(x) = cos(2x+1) d/dx(2x+1)

⇒cos(2x+1)(2)

⇒2 cos(2x+1)

Derivative of Sin 2x Practice Questions

Q1: The derivative of sin 3x is?

Q2: The derivative of sin²x ?

Q3: What is the derivative of sin nx?

Q4: Find the derivative of sin2x cosx?

Derivative of Sin 2x Frequently Asked Questions

What is Meaning of sin 2x?

Sin 2x is one of the formulae for double angles in trigonometry. Using this formula, we can get the sine of the doubled value of an angle. Thus sin 2x = 2 sin x cos x.

What is Formula of sin 2x?

The formula for sin 2x is 2 sin x.cos x

What is Derivative of sin 2theta?

Derivative of sin 2theta is 2 Cos (2theta)

When is Sin 2x zero?

The sine function is equal to 0 only at certain angles, specifically at angles that are multiples of π (pi), such as 0, π, 2π, 3π, etc. So, the solutions of the equation sin(2x) = 0 are the angles where 2x is an integer multiple of π.