Derivative of Cos 2x

Learn what is the derivative of cos 2x and understand how to prove the derivative of cos 2x by first principle, chain rule and quotient rule.

Alan Walker-

Published on 2023-05-26

Introduction to differentiation of cos2x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of cos(2x) can be calculated by following the rules of differentiation. Or, we can directly find the derivative of cos 2x by applying the first principle of differentiation. In this article, you will learn what the differential of cos^2x is and how to calculate the cos^2x derivative by using different approaches.

What is the derivative of cos(2x)?

The derivative of cos 2x with respect to x is -2sin 2x, which tells us how fast the cosine function is changing. We write it as d/dx(cos(2x)). In a triangle, the cosine function represents the ratio of the length of the adjacent side to the length of the hypotenuse. We write it as:
$\cos x = \frac{\text{base}}{\text{hypotenuse}}$
Understanding the derivative of cos 2x is useful in various applications, such as physics and finance. By knowing the rate of change of the cosine function, we can analyze the behavior of various systems and make informed decisions. Also we can find the derivative of cos 3x just by replacing 2 with 3 in the derivative of cos 2x.

Derivative of cos 2x formula

The formula for finding the derivative of cos(2x) is simple! It is the negative sine function of 2x, which can be written as;

$\frac{d}{dx}(\cos 2x) \;=\; -2\sin 2x $

This formula allows us to determine how the cos(2x) function changes over an independent variable 'x'. Further, if the value of x is known, then we can calculate the rate of change in cos 2x at that point. Remember that the negative sign in the formula means that the slope or rat of change of the cos(2x) function is going down, or decreasing, at the point you are calculating the derivative for. 

How do you prove the derivative of cos2x?

There are different ways or methods to derive derivatives of cos 2x. These methods allow you to find the rate of change or slope of the cos(2x) function at any point, which is useful in many applications. We can prove the differential of cos^2x by using the following derivative rules;

  1. First Principle
  2. Chain Rule
  3. Quotient Rule

Derivative of cos2x by first principle

The derivative first principle tells that the differentiation cos 2x is equal to the negative of 2sin x. The derivative of a function by first principle refers to finding the slope of a curve by using algebra. It is also known as the delta method. Mathematically, the first principle of derivative formula is represented as:

$f(x) \;=\; \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $

Proof of derivative of cos 2x by first principle

To prove the derivative of cos (2x) by using first principle, replace f(x) by cos(2x). Or, if we want to calculate derivative of cos(3x), then replace f(x) by cos(3x).

$f(x) \;=\; \lim_{h \to 0} \frac{\cos 2(x+h) -\cos 2x}{h} $

Now, by using trigonometric formula,

$f(x) \;=\; \lim_{h \to 0}-2\frac{\sin\frac{2(x+h)+2x}{2} \sin\frac{2(x+h)-2x}{2}}{h}$


$f(x) \;=\; \lim{h \to 0} \;-2\frac{\sin(2x+h)\sin h}{h}$

As h approaches to zero, [sin h / h] becomes 1. So,

$f(x) \;=\; -2sin 2x$

Hence the differentiation of cos2x is equal to -2sin 2x. Also use our first derivative calculator to find first derivative of a function in a few steps.

Derivative of cos 2x by chain rule

To calculate the derivative of cos(2x), we can use the chain rule since the cosine function can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. In mathematical terms, it can be expressed as:

$\frac{dy}{dx} \;=\; \frac{dy}{du} \times \frac{du}{dx}$

If you need instant and accurate results, you can use chain rule multivariable calculator It's a convenient tool that can simplify your calculations and save you time.

Proof of cos2x differentiation by chain rule

To prove the derivative of cos 2x by using chain rule, we first assume that,

$y \;=\;\cos u \;\; and \;\; u \;=\; 2x$

Now finding the derivatives of both functions such as;

$\frac{dy}{du} \;=\; -\sin u \;\; and \;\; \frac{du}{dx} \;=\;2$

Now using these in chain rule formula,

$\frac{dy}{dx} \;=\; \frac{dy}{du} \times \frac{du}{dx}$

$\frac{dy}{dx} \;=\; -\sin u \times 2$

Using the value of u,

$\frac{dy}{dx} \;=\; -2\sin 2x$

Thus, we have proven that the derivative of cos squared x using the chain rule is always equal to -2sin 2x.

Derivative of cos 2x using quotient rule

Since the tangent is the ratio of two trigonometric ratios sine and cosine. Therefore, the derivative of tangent can also be calculated by using the quotient rule. The quotient rule is defined as;

$\frac{d}{dx}(\frac{f(x)}{g(x)}) \;=\; \frac{f(x).g(x)-g(x).f(x)}{(g(x))^2}$

Proof of derivative of cos2x by quotient rule

To prove the derivative of cosine, we can use the quotient rule. We begin by expressing the cosine of 2x as the raciprocal of the secant of 2x such as;

$\frac{d}{dx}(\cos 2x) \;=\; \frac{d}{dx} (\frac{1}{\sec 2x}) $

Applying the quotient rule,

$\frac{d}{dx}(\cos 2x) \;=\; (\frac{0.(\sec 2x)-2\sec 2x\tan 2x}{(\sec 2x )^2})$

$\frac{d}{dx}(\cos 2x) \;=\; (\frac{-2\sec 2x\tan 2x}{(\sec 2x )^2})$

$\frac{d}{dx}(\cos 2x) \;=\; \frac{-2tan 2x}{\sec 2x}$


$\tan x \;=\; \frac{\sin x}{\cos x} \;\; and \;\; \frac{1}{\sec x} \;=\; \cos x$


$\frac{d}{dx}(\cos 2x) \;=\; -\frac{2\sin 2x}{\cos 2x} \cos 2x \;=\; -2\sin 2x$

Therefore, we have successfully derived the derivative of cos 2x using the quotient rule of differentiation.

Related: Use derivative quotient rule calculator for learning and practicing online.

How to find the derivative of cos 2x with a calculator?

The easiest way to calculate the derivative of cos 2x is by using an online tool. You can use our derivative solver for this. Here are the steps to calculate the derivative of cos 2x using this tool:

  1. Write the function as cos 2x in the 'enter function' box. In this step, you need to provide input value as a function as you have to calculate the derivative of cos 2x.

  2. Now, select the variable by which you want to differentiate cos 2x. Here you have to choose 'x'.

  3. Select how many times you want to differentiate cosine2 x. In this step, you can choose 2 for second, 3 for third derivative and so on.

  4. Click on the calculate button.

After completing these steps, you will receive the derivative of cos 2x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

Frequntly Asked Questions

What is the derivative of cos2x by first principle?

The derivative of cos 2x can be derived by using different techniques of derivatives. But it will be the same whatever technique you use to calculate derivatives. Hence the derivative of cos^2x by first principle is -2sin 2x.

What is the derivative of Cos3x?

The derivative of a cos 3x is similar to the derivative of cos x. There will be only one difference of 3. It is calculated as;

$\frac{d}{dx}(\cos 3x) =-3\sin 3x$

What is cos function?

In a triangle, the cos function is the ratio of adjacent to hypotenuse. It is written as;

$\cos x = \frac{Base}{Hypotenuse}$

Where, the base is the horizontal side and hypotenuse is the longest side of a triangle.

What is the derivative of cos 2x?

The derivative of y = cos 2x is;

$\frac{dy}{dx} = -2\sin 2x$

Or, we can calculate it by using chain rule also.

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