Derivative of Sin 2x

Before going to find the derivative of sin 2x, let us recall a few facts about sin 2x. 2x is a double angle and by one of the double angle formulas, sin 2x = 2 sin x cos x. Since sin 2x involves double angle, its derivative also involves the double angle. In this article, we are going to prove that the differentiation of sin 2x is 2 cos 2x using various methods.

Note that sin 2x and sin²x are different from each other. In this article, we are going to see the difference between the derivatives of sin 2x and sin square x.

The derivative of sin 2x is 2 cos 2x. We write this mathematically as d/dx (sin 2x) = 2 cos 2x (or) (sin 2x)' = 2 cos 2x. Here, f(x) = sin 2x is the sine function with double angle. We can do the differentiation of sin 2x in different methods such as:

• Using the first principle
• Using the chain rule
• Using product rule

Derivative of Sin 2x Formula

The derivative of sin 2x is 2 cos 2x. It can be written as

• d/dx (sin 2x) = 2 cos 2x
• (sin 2x)' = 2 cos 2x

Let us prove this in different methods as mentioned above.

Here is the differentiation of sin 2x by the first principle. For this, let us assume that f(x) = sin 2x. Then f(x + h) = sin 2(x + h) = sin (2x + 2h). Substituting these values in the formula of the derivative using first principle ( the limit definition of the derivative),

f'(x) = lim_h→0 [f(x + h) - f(x)] / h

f'(x) = lim_h→0 [sin (2x + 2h) - sin 2x] / h

We can simplify this limit in two methods.

Method 1

By one of the trigonometric formulas, sin C - sin D = 2 cos [(C + D)/2] sin [(C - D)/2]. Applying this,

f'(x) = lim_h→0 [2 cos[(2x + 2h + 2x)/2] sin[(2x +2h - 2x)/2] ] / h

= lim_h→0 [2 cos[(4x + 2h)/2] sin (h) ] / h

= 2 lim_h→0 [cos[(4x + 2h)/2] · lim_h→0 [sin h / h]

Using limit formulas, lim_x→0 (sin x/x) = 1. So

f'(x) = 2 [cos[(4x + 0)/2] · (1) = 2 cos (4x/2) = 2 cos 2x

Thus, we have proved that the derivative of sin 2x is 2 cos 2x.

Method 2

By sum and difference formulas,

sin (A + B) = sin A cos B + cos A sin B

Using this,

f'(x) = lim_h→0 [sin 2x cos 2h + cos 2x sin 2h - sin 2x] / h

= lim_h→0 [ - sin 2x (1- cos 2h) + cos 2x sin 2h] / h

= lim_h→0 [ - sin 2x (1 - cos 2h)]/h + lim_h→0 (cos 2x sin 2h)/h

= -sin 2x lim_h→0 (1 - cos 2h)/h + (cos 2x) lim_h→0 sin 2h/h

Using double angle formulas, 1 - cos(2h) = 2 sin²(h).

f'(x) = -sin 2x lim_h→0 (2 sin²(h))/h + (cos 2x) lim_h→0 sin 2h/h

= -2sin 2x [lim_h→0 sin (h) / (h) · lim_h→0 sin h] + (cos 2x) (lim_h→0 sin 2h/h)

By one of the limit formulas, lim_x→0 (sin x/x) = 1 and lim_x→0 (sin ax/x) = a. So

f'(x) = -2sin 2x (1 · sin 0) + cos 2x (2)

= -2sin 2x(0) + 2 cos 2x (From trigonometric table, sin 0 = 0)

= 2 cos 2x

Hence we have derived that the derivative of sin 2x is 2 cos 2x.

Thus, the derivative of sin 2x is found by the first principle.

We can do the differentiation of sin 2x using the chain rule because sin 2x can be expressed as a composite function. i.e., we can write sin 2x = f(g(x)) where f(x) = sin x and g(x) = 2x (one can easily verify that f(g(x)) = sin 2x). We know that the derivative of sin x is cos x.

Then f '(x) = cos x and g'(x) = 2. By chain rule, the derivative of f(g(x)) is f '(g(x)) · g'(x). Using this,

d/dx (sin 2x) = f '(g(x)) · g'(x)

= f '(2x) · (2)

= cos 2x (2)

= 2cos 2x

Thus, the derivative of sin 2x is found by using the chain rule.

To find the derivative of f(x) = sin 2x by the product rule, we have to express sin 2x as the product of two functions. Using the double angle formula of sin, sin 2x = 2 sin x cos x. Let us assume that u = 2 sin x and v = cos x. Then u' = 2 cos x and v' = -sin x. By product rule,

f '(x) = uv' + vu'

= (2 sin x) (- sin x) + (cos x) (2 cos x)

= 2 (cos²x - sin²x)

= 2 cos 2x

This is because, by the double angle formula of cos, cos 2x = cos²x - sin²x.

Thus, we have found the derivative of sin 2x by using the product rule.

n^th Derivative of Sin 2x

n^th derivative of sin 2x is the derivative of sin 2x that is obtained by differentiating sin 2x repeatedly for n times. To find the n^th derivative of sin 2x, let us find the first derivative, the second derivative, ... up to a few times to understand the trend/pattern.

• 1^st derivative of sin 2x is 2 cos 2x
• 2^nd derivative of sin 2x is -4 sin 2x
• 3^rd derivative of sin 2x is -8 cos 2x
• 4^th derivative of sin 2x is 16 sin 2x
• 5^th derivative of sin 2x is 32 cos 2x
• 6^th derivative of sin 2x is -64 sin 2x
• 7^th derivative of sin 2x is -128 cos 2x
• 8^th derivative of sin 2x is 256 sin 2x and so on.

Using this trend, we can define the n^th derivative of sin 2x as follows:

• (sin 2x)⁽ⁿ⁾is:
  2ⁿ sin 2x, if n is a multiple of 4,
  -2ⁿ cos 2x, if n is 1 less than a multiple of 4,
  -2ⁿ sin 2x, if n is 2 less than a multiple of 4,
  2ⁿ cos 2x, if n is 3 less than a multiple of 4,

This can be written in another way as follows:

The derivative of sin²x is NOT the same as the derivative of sin 2x. The differentiation of sin square x is sin 2x. Let us see how. Let f(x) = sin²x. This can be written as f(x) = (sin x)². To find its derivative, we can use a combination of the power rule and the chain rule. Then we get,

f'(x) = 2(sin x) d/dx(sin x)

= 2 sin x cos x

= sin 2x (by using the double angle formula of sin)

Therefore, the derivative of sin²x is sin 2x.

Important Notes on Derivative of Sin 2x:

• The differentiation of sin 2x is 2 cos 2x.
• In general, the derivative of sin ax is a cos ax.
  For example, the derivative of sin (-3x) is -3 cos(-3x), the derivative of sin 5x is 5 cos 5x, etc.
• The derivatives of sin 2x and sin²x are NOT the same.
  d/dx (sin 2x) = 2 cos 2x
  d/dx (sin²x) = sin 2x

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What is Derivative of Sin 2x?

The derivative of sin 2x is 2 cos 2x. Mathematically, it is written as d/dx(sin 2x) = 2 cos 2x (or) (sin 2x)' = 2 cos 2x.

What is the Difference Between the Derivative and Anti Derivative of Sin 2x?

The derivative of sin 2x is cos 2x. The antiderivative of sin 2x is nothing but its integral and the integral of sin 2x is (-1/2) cos 2x + C. We can find this integral by using the substitution 2x = u.

How to Find the Derivative of Sin 2x Using Chain Rule?

Sin 2x can be written as the composite function f(g(x)) where f(x) = sin x and g(x) = 2x. So by chain rule, d/dx (sin 2x) = f'(g(x)) g'(x) = cos (2x) (2) = 2 cos 2x.

What is the Double Derivative of Sin Square x?

The first derivative of sin²x is 2 sin x cos x (or) sin 2x. The second differentiation of sin square x is d/dx(sin 2x) = 2 cos 2x.

What is the Derivative of Sin 2x With Respect to Cos 2x?

Let u = sin 2x and v = cos 2x. We have to find du/dv · du/dx = 2 cos 2x and dv/dx = -2sin 2x. We have du/dv = (du/dx) / (dv/dx) = (2 cos 2x) / (-2 sin 2x) = - cot 2x.

What is the Limit Definition of the Derivative of Sin 2x?

Using the limit definition of the derivative, the derivative of f(x) is f'(x) = lim_h→0 [f(x + h) - f(x)] / h. Thus, the limit definition of the derivative of f(x) = sin 2x is, lim_h→0 [sin (2x + 2h) - sin 2x] / h. To see how to evaluate this limit, click here.

How to Find the Derivative of Sin 2x Using Product Rule?

We can write sin 2x as 2 sin x cos x. Taking derivative, d/dx(sin 2x) = d/dx (2 sin x cos x) = (2 sin x) d/dx (cos x) + (cos x) d/dx (2 sin x) = (2 sin x) (- sin x) + (cos x) (2 cos x) = 2 (cos²x - sin²x) = 2 cos 2x. Thus, the derivative of sin 2x is 2 cos 2x.

What is the Second Derivative of Sin 2x?

The first differentiation of sin 2x with respect to x is 2 cos 2x. Differentiating it again we can get its second derivative to be -4 sin 2x.