Derivative of Sin 2x First Principle, Chain Rule & Product Rule Explained

The derivative tells us how a function changes as its input changes — it shows the rate of change. In calculus, derivatives are important for solving many types of problems, especially those involving curves and motion.

Now, when we look at the function sin(2x), the "2x" means the angle is doubled — this is called a double angle. From trigonometry, we know that sin(2x) = 2 sin(x) cos(x), which is one of the double angle formulas.

In this topic, we aim to find the derivative of sin(2x) using different methods such as the first principle, the chain rule, and the product rule. No matter which method we choose, the result will show that the derivative of sin(2x) is 2 cos(2x). We’ll break down the steps clearly so it’s easy to understand, even if you’re just starting with calculus.

What is the Derivative of Sin 2x?

The derivative of sin 2x is 2 cos 2x. It shows how the function sin 2x changes with respect to x. In symbols, it is written as:

d/dx (sin 2x) = 2 cos 2x
or
(sin 2x)’ = 2 cos 2x

Since 2x is a double angle, its derivative also reflects that by giving 2 cos 2x. This result can be found using different methods like the chain rule or first principle. In this topic, we will explore these methods to clearly understand why the derivative of sin 2x is 2 cos 2x.

We can differentiate sin 2x using various methods such as:

1. First principle of Derivatives
2. Chain rule
3. Product rule

The first principle is used to differentiate . Assume that this.

Consequently,

. Using the first principle, substitute these values in the derivative formula.

Using the sum and difference formulas

Step 1 :
Step 2 :
Step 3 :
Step 4 :
Step 5 :

Applying double angle formula

Step 1 :

Step 2:

Apply limit formulas

Step 1 :

Step 2 :

Step 3 :

As a result, we can conclude that the derivative of is

Learn about Derivative of Cos3x

Derivative of Sin 2x Proof by Chain Rule

The chain rule is a calculus formula that computes the derivative of a function combination of two or more functions.

We know that the chain rule is,

Now, applying it to the given derivative, we get

Hence proved.

Derivative of Sin 2x Proof by Product Rule

We must express sin 2x as the product of two functions in order to find the derivative of using the product rule. Using the sin double angle formula, Sin2x = 2 sinx cosx

Assume that,

Step 1 :

Step 2 :

Then,

Step 3:

Step 4 :

According to product rule:

Hence, proved.

Nth Derivative of Sin 2x

The nth derivative of sin 2x is the derivative of sin 2x obtained by differentiating sin 2x n times. To find the nth derivative of sin 2x, first find the first derivative, then the second derivative, and so on until the trend/pattern is understood.

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

Consider the product of these functions’ derivatives.

Step 1 :

Now, n^th Derivative of Sin 2x

Step 2 :

Where, indicates the number of i -combinations of elements.

Important Notes on Derivative of Sin 2x:

The derivative (or differentiation) of sin 2x is 2 cos 2x.

In general, if you have a function like sin(ax), its derivative will be a cos(ax).
For example:

• The derivative of sin(5x) is 5 cos(5x).
• The derivative of sin(-3x) is -3 cos(-3x).

It’s important to know the difference between sin 2x and sin²x:

• The derivative of sin 2x is 2 cos 2x.
• But the derivative of sin²x (that is, sin x raised to the power 2) is 2 sin x · cos x, not the same as sin 2x.

Derivative of Sin2x Solved Examples

Example 1: Find the derivative of sin(3x + 2)

Solution:

Let us take,
f(x) = sin(3x + 2)

To differentiate this, we use the chain rule.

Step 1: The outer function is sin(u), and its derivative is cos(u).
Step 2: The inner function is (3x + 2), and its derivative is 3.

Now apply the chain rule:
f'(x) = cos(3x + 2) × 3
= 3 cos(3x + 2)

The derivative of sin(3x + 2) is 3 cos(3x + 2)

Example 2: Find the derivative of sin(3x) × cos(x)

Solution:

Let f(x) = sin(3x) × cos(x)

To solve this, use the product rule:
If f(x) = u × v, then f'(x) = u'v + uv'

Here,
u = sin(3x), and v = cos(x)

Step 1: Derivative of sin(3x) is 3 cos(3x) (using chain rule)
Step 2: Derivative of cos(x) is -sin(x)

Now apply the product rule:
f'(x) = [sin(3x)] × [-sin(x)] + [cos(x)] × [3 cos(3x)]
= -sin(3x) sin(x) + 3 cos(3x) cos(x)

The derivative of sin(3x) × cos(x) is 3 cos(3x) cos(x) – sin(3x) sin(x)

Example 3: Find the derivative of sin(4x) × cos(x)

Solution:

Let f(x) = sin(4x) × cos(x)

Again, use the product rule:

Step 1: Derivative of sin(4x) is 4 cos(4x)
Step 2: Derivative of cos(x) is -sin(x)

Now apply the product rule:
f'(x) = sin(4x) × (-sin(x)) + cos(x) × (4 cos(4x))
= -sin(4x) sin(x) + 4 cos(4x) cos(x)

The derivative of sin(4x) × cos(x) is 4 cos(4x) cos(x) – sin(4x) sin(x)

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If you are checking Derivative of sin2x article, also check related maths articles:
Derivative of x	Derivatives of Logarithmic Functions
Differential Calculus	Applications of Derivatives
Derivative of cot x	Derivative of Sin3x

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