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 derivative of sin 2x to be 2 cos 2x using various methods.
Let us derive the derivative of sin 2x in different methods and solve a few examples using the same. Also, let us see the difference between the derivatives of sin 2x and sin^{2}x.
What is the Derivative of Sin 2x?
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.
Derivative of Sin 2x Proof by First Principle
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) = limₕ→₀ [f(x + h)  f(x)] / h
f'(x) = limₕ→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) = limₕ→₀ [2 cos[(2x + 2h + 2x)/2] sin[(2x +2h  2x)/2] ] / h
= limₕ→₀ [2 cos[(4x + 2h)/2] sin (h) ] / h
= 2 limₕ→₀ [cos[(4x + 2h)/2] · limₕ→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) = limₕ→₀ [sin 2x cos 2h + cos 2x sin 2h  sin 2x] / h
= limₕ→₀ [  sin 2x (1 cos 2h) + cos 2x sin 2h] / h
= limₕ→₀ [  sin 2x (1  cos 2h)]/h + limₕ→₀ (cos 2x sin 2h)/h
= sin 2x limₕ→₀ (1  cos 2h)/h + (cos 2x) limₕ→₀ sin 2h/h
Using double angle formulas, 1  cos(2h) = 2 sin^{2}(h).
f'(x) = sin 2x limₕ→₀ (2 sin^{2}(h))/h + (cos 2x) limₕ→₀ sin 2h/h
= 2sin 2x [limₕ→₀ sin (h) / (h) · limₕ→₀ sin h] + (cos 2x) (limₕ→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.
Derivative of Sin 2x Proof by Chain Rule
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.
Derivative of Sin 2x Proof by Product 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 (cos^{2}x  sin^{2}x)
= 2 cos 2x
This is because, by the double angle formula of cos, cos 2x = cos^{2}x  sin^{2}x.
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:
2^{n} sin 2x, if n is a multiple of 4,
2^{n} cos 2x, if n is 1 less than a multiple of 4,
2^{n} sin 2x, if n is 2 less than a multiple of 4,
2^{n} cos 2x, if n is 3 less than a multiple of 4,
This can be written in another way as follows:
Derivative of Sin^2 x
The derivative of sin^{2}x is NOT the same as the derivative of sin 2x. The derivative of sin^{2}x is sin 2x. Let us see how. Let f(x) = sin^{2}x. 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 sin^{2}x is sin 2x.
Important Notes on Derivative of Sin 2x:
 The derivative 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 sin^{2}x are NOT the same.
d/dx (sin 2x) = 2 cos 2x
d/dx (sin^{2}x) = sin 2x
Topics Related to Derivative of Sin 2x:
Examples Using Derivative of Sin 2x

Example 1: Find the derivative of sin (2x + 1).
Solution:
Let f(x) = sin (2x + 1)
Using the chain rule,
f'(x) = cos(2x + 1)^{ }d/dx (2x + 1)
= cos(2x + 1)^{ }(2)
= 2 cos(2x+1)
Answer: The derivative of sin (2x + 1)^{ }is 2 cos(2x + 1).

Example 2: What is the derivative of sin^{2}x cos x?
Solution:
Let f(x) = sin^{2}x cos x
By product rule,
f'(x) = sin^{2}x d/dx (cos x) + (cos x) d/dx (sin^{2}x)
= sin^{2}x ( sin x) + (cos x) (sin 2x)
= sin 2x cos x  sin^{3}x
Answer: The derivative of sin^{2}x cos x is 2 (sin 2x cos x  sin^{3}x).

Example 3: What is the derivative of sin 2x cos x.
Solution:
Let f(x) = sin 2x cos x.
Using the product rule,
f'(x) = sin 2x d/dx (cos x) + cos x d/dx (sin 2x)
= sin 2x (sin x) + sin x (2 cos 2x)
= 2 cos 2x sin x  sin x sin 2x
Answer: The derivative of sin 2x cos x is 2 cos 2x sin x  sin x sin 2x.
FAQs on Derivative of Sin 2x
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^2x?
The first derivative of sin^{2}x is 2 sin x cos x (or) sin 2x. The second derivative of sin^{2}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 derivative, the derivative of f(x) is f'(x) = limₕ→₀ [f(x + h)  f(x)] / h. Thus, the limit definition of the derivative of f(x) = sin 2x is, limₕ→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 (cos^{2}x  sin^{2}x) = 2 cos 2x. Thus, the derivative of sin 2x is 2 cos 2x.
What is the Second Derivative of Sin 2x?
The first derivative 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.
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