Derivative of Cos 2x
Derivative of cos 2x is 2 sin 2x which is the process of differentiation of the trigonometric function cos 2x w.r.t. angle x. It gives the rate of change in cos 2x with respect to angle x. The derivative of cos 2x can be derived using different methods. Mathematically, the derivative of cos 2x is written as d(cos 2x)/dx = (cos 2x)' = 2sin 2x.
In this article, we will prove the derivative of cos 2x using different methods including the first principle of differentiation and chain rule. We will also compare the graphs of the trigonometric function cos 2x and the derivative of cos 2x along with some examples.
What is Derivative of Cos 2x?
The derivative of cos 2x is negative of twice the trigonometric function sin 2x, that is, 2 sin 2x. The derivative of cos 2x is denoted as d(cos 2x)/dx or (cos 2x)'. To derive the derivative of cos 2x, different trigonometric formulas and identities are used along with some rules of differentiation. It can be derived using the definition of the limits, and chain rule. Since the derivative of cos 2x is 2 sin 2x, therefore the graph of the derivative of cos 2x will be the graph of the negative of 2 sin 2x.
Derivative of Cos 2x Formula
Now, we will write the derivative of cos 2x mathematically. The formula for the derivative of cos 2x is:
 d(cos 2x)/dx = 2 sin 2x
 (cos 2x)' = 2 sin 2x
Derivative of Cos 2x Using Chain Rule
The chain rule method for determining the derivatives of functions is: (f(g(x)))’ = f’(g(x)) . g’(x). Now, to evaluate the derivative of cos 2x using the chain rule, we will use certain algebraic and trigonometric properties and identities such as:
 d(cos x)/dx = sin x
 d(ax)/dx = a, where a is a real number
We can write the derivative of cos 2x with respect to x as a product of the derivative of cos 2x with respect to 2x and the derivative of 2x with respect to x, that is, d(cos 2x)/dx = d(cos 2x)/d(2x) × d(2x)/dx. Now, using the above formulas and chain rule, we have,
d(cos 2x)/dx = d(cos 2x)/d(2x) × d(2x)/dx
= sin 2x × 2
= 2 sin 2x
Hence, we have derived the derivative of cos 2x as 2 sin 2x using the chain rule.
Derivative of Cos 2x Using the First Principle of Differentiation
Now, we will prove that the derivative of cos 2x is  2 sin 2x using the definition of limits, that is, the first principle of derivatives. To find the derivative of cos 2x, we take the limiting value as x approaches x + h. To simplify this, we set x = x + h, and we want to take the limiting value as h approaches 0. We are going to use certain differentiation and trigonometry formulas to determine the derivative of cos 2x. The formulas are:
 cos A  cos B = 2 sin[(A + B)/2] sin[(A  B)/2]
 \(f'(x)=\lim_{h\rightarrow 0}\dfrac{f(x+h)f(x)}{h}\)
 \(\lim_{x\rightarrow 0} \dfrac{\sin x}{x} = 1\)
Using the above formulas, we have
\(\begin{align} \frac{\mathrm{d} \cos 2x}{\mathrm{d} x}&=\lim_{h\rightarrow 0}\dfrac{\cos 2(x+h)\cos (2x)}{h}\\&=\lim_{h\rightarrow 0}2\dfrac{\sin (\frac{2(x+h)+2x}{2})\sin (\frac{2(x+h)2x}{2})}{h}\\&=\lim_{h\rightarrow 0}2\dfrac{\sin (2x+h)\sin (h)}{h}\\&=2\lim_{h\rightarrow 0}\sin (2x+h)\times \lim_{h\rightarrow 0}\dfrac{\sin h}{h}\\&=2 \sin 2x \times 1\\&=2 \sin 2x\end{align}\)
Hence we have derived the derivative of cos 2x using the first principle of differentiation.
Graph of Derivative of Cos 2x
We know that the derivative of cos 2x is negative of twice the trigonometric function sin 2x which implies the graph of the derivative of cos 2x is similar to the graph of the trigonometric function sin 2x with negative values where sin 2x has positive values. First, let us see how the graphs of cos 2x and the derivative of cos 2x looks like. As sin 2x is a periodic function, the graph of differentiation of cos 2x is also periodic and its period is π.
AntiDerivative of Cos 2x
As the name suggests, antiderivative is the inverse process of differentiation. The antiderivative of cos 2x is nothing but the integral of cos 2x. We know that the integral of cos x is sin x + C. Using the formula of integration ∫cos(ax + b) = (1/a) sin(ax + b) + C, the antiderivative of cos 2x is (1/2) sin 2x + C, where C is constant of integration. Hence, we have obtained the antiderivative of cos 2x as (1/2) sin 2x + C.
∫cos 2x = (1/2) sin 2x + C
Important Notes on Derivative of Cos 2x
 Derivative of cos 2x is NOT equal to sin 2x. We use the chain rule to determine the derivative of cos 2x.
 Derivative of cos 2x can also be determined using the cos 2x formula.
 The derivatives of cos 2x and cos^{1}2x are NOT the same.
Topics Related to Derivative of Cos 2x
Examples Using Derivative of Cos 2x

Examples 1: Find the derivative of cos 2x using the cos 2x formula.
Solution: To determine the derivative of cos 2x using the cos 2x formula, we will use the chain rule of differentiation and trigonometric identitities:
 cos 2x = cos^{2}x  sin^{2}x
 d(cos x)/dx = sin x
 d(sin x)/dx = cos x
 sin 2x = 2 sin x cos x
d(cos 2x)/dx = d(cos^{2}x  sin^{2}x)/dx
= d(cos^{2}x)/dx  d(sin^{2}x)/dx
= 2 × cos x × (sin x)  2 × sin x × (cos x)
= 4 cos x sin x
= 2 sin 2x

Example 2: Determine the derivative of negative of cos 2x using the derivative of cos 2x.
Solution: Derivative of cos 2x is 2 sin 2x.
d(cos 2x)/dx = d(cos 2x)/dx
=  (2 sin 2x)
= 2 sin 2x
Answer: d(cos 2x)/dx = 2 sin 2x
FAQs on Derivative of Cos 2x
What is Derivative of Cos 2x in Trigonometry?
The derivative of cos 2x is negative of twice the trigonometric function sin 2x, that is, 2 sin 2x.
How to Find the Derivative of Cos 2x?
We can find the derivative of cos 2x using different methods including the first principle of differentiation, cos 2x formula and chain rule.
What is the Derivative of 1 + cos 2x?
The derivative of 1 + cos 2x is given by d(1 + cos 2x)/dx = 0  2sin 2x = 2 sin 2x. Therefore, the derivative of 1 + cos 2x is equal to the derivative of cos 2x.
What is the Antiderivative of cos 2x?
The antiderivative of cos 2x is (1/2) sin 2x + C, where C is the constant of integration.
Is the Derivative of Cos 2x the Same as the Derivative of Cos^{1}2x?
No, the Derivative of Cos 2x is Not the Same as the Derivative of Cos^{1}2x.
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