Derivative of Cos

Learn about derivative of cos, formula, proof, solution and how to calculate derivatives of cos x with chain and product rules.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of cosx

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

What is the derivative of cos?

The derivative cos, with respect to the variable x, is equivalent to the negative sine of x. Symbolically, it is represented as d/dx(cos x). This derivative represents the rate of change of the trigonometric function cos x. In a triangle, cosine is defined as the ratio of the adjacent side to the hypotenuse. Mathematically, it is expressed as:

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

Therefore, the differential of cos x represents the rate of change of the trigonometric function cosine with respect to the variable x.

Derivative of cos x formula

The formula for finding the derivative of the cosine function, denoted as d/dx(cos x), is equal to the negative of the sine function of x. Mathematically, the cos differentiation formula can be written as:

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

This formula allows us to determine how the cos x function changes over an independent variable 'x'. 

How do you prove the derivative of cosx?

There are various ways to prove the cos x derivative. These are:

  1. First Principle

  2. Chain Rule

  3. Quotient Rule

Each method provids a different way to compute the differentiation of cos function. By using these methods, we can mathematically prove the formula for finding differential of cos x.

Derivative of cosine by first principle

According to the first principle of derivative the cos derivative is equal to the negative of sin x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

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

This formula allow us to determine the rate of change of a function at a specific point by using limit definition of derivative.

Proof of derivative of cos x by first principle

To prove the differential of cos x by using first principle, replace f(x) by cos x in the general formula for the first principle.

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

we can also replace f(x) by cos2x to calculate the cos2x derivative

Using the trigonometric identity cos(x+h) = cos x cos h - sin x sin h, we can simplify the formula as follows:

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


$f(x) \;=\; \lim_{h \to 0} \frac{\cos x (\cos h - 1) - \sin h \sin x}{h}$

When h approaches zero, sin h / h becomes 1. Therefore, we can rewrite the formula as follows:

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

Hence the differentiation of cos is equal to -sin x. You can also use our derivative by definition calculator online to find the differential of cos easily.

Cos derivative formula by chain rule

To calculate the cos x differentation, 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:

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

Where, u=g(x), y=f(u), dy/du is the derivative of f(u) with respect to u and du/dx is the derivative of g(x) with respect to x.

Proof of derivative of cos x by chain rule

To prove the differential of cos x by using chain rule, we will use the following trigonometric ratios.

  • $cos (\frac{π}{2}-\theta) \;=\; \sin Θ $

  • $sin (\frac{π}{2}-\theta) \;=\; \cos Θ $

Using the above trigonometric identities, we can write the cos derivative such as;

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

That is;

$\frac{d}{dx}(\cos x) \;=\; \cos(\frac{π}{2}-Θ)(-1)) $


$\cos(\frac{π}{2}-Θ) \;=\; \sin Θ $


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

Hence the chain rule also proved that the differentiation of cos is always -sin x. You can also use chain rule calculator, because it can be written as the combination of two funcitons.

Derivative of cos x 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;


Proof of derivative cosine by quotient rule

To prove the cosine derivative by using the quotient rule formula, we can write it as,

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

By using quotient rule,

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

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

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


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


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

Hence, we have derived the cos derivative formula using the quotient rule derivative calculator.

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

The easiest way to calculate the cos differentiation formula, is by using an online dy/dx calculator. You can use our derivative calculator for this. Here, we provide you a step-by-step way to find differential of cos proof by using this tool.

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

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

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

  4. Click on the calculate button. After this step, you will get the derivative cosine x within a few seconds.

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

Frequently Asked Questions

What is the derivative of cos and sin?

The derivative cos formula is negative of sin such as sin x. And the derivative of sinx is cosx. The derivative of both functions can be easily calculated by applying the rules of differentiation.

What is the derivative of 1/Cos?

The derivative of a secant or 1/cos x is the product of tan x and sec x. It is calculated as;

$ \frac{d}{dx}(\frac{1}{\cos x}) \;=\; \frac{0.\cos x- 1.(-\sin x)}{(\cos x ^2)} $

$ \frac{d}{dx}(\frac{1}{\cos x}) \;=\; \frac{-\sin x}{\cos x^2}$

Using trigonometric ratios,

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

We have,

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

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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