Derivative of cscx

Learn what is the derivative of csc x with formula and proof. Also understand how to prove the derivative of cscx by using quotient and product rule.

Alan Walker-

Published on 2023-05-26

Introduction to the derivative of cscx

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

What is the derivative of cscx?

The derivative of csc or cosecant x is a commonly searched topic in calculus. The derivative of cscx with respect to x can be found using the formula d/dx(csc x) = -csc x.cot x, which involves taking the derivative of both the sine and cosine functions. This formula can be useful for finding the rate of change of csc x, which is the reciprocal of the sine function, in various applications. In a right triangle, the cosecant x is equal to the ratio of the hypotenuse to the opposite side, or 1/sin x. By using this relationship, we can find the value of cosecant x for any given angle.

Derivative of csc(x) formula

The derivative of cscx, also known as cosecant x, can be found using the formula:

d/dx (csc x) = -cot x . csc x.

This formula can be useful in finding the rate of change of the cosecant function, which is the reciprocal of the sine function. Remember that the derivative of -cscx is the same as the cscx derivative. Using this formula, you can easily find the derivative of csc x or solve problems like finding the second derivative f(x) = csc(x).

How to calculate csc derivative?

There are mulitple derivative rules 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 csc x.

Derivative of cscx by first principle

A fundamental way to find the derivative of a function is by using the first principle, which is also known as the delta method. This method involves finding a general expression for the slope of a curve by using algebra. The derivative is a measure of the instantaneous rate of change of a function at a specific point, and it can be calculated using the limit formula:

f'(x)=lim f(x+h)-f(x)/h

This formula represents the slope of the tangent line to the curve of the function at the point x. Understanding the first principle can be helpful in finding the derivative of csc x or other functions, especially when other methods like the chain rule or quotient rule are not applicable.

Proof of cosecant derivative by first principle

To prove the derivative of csc(x) by using first principle, replace f(x) by csc x or replace by csc 2x to calculate derivative of csc2x. f(x)=limh→0f(x+h)-f(x)/h

f'(x) = lim cosec (x+h) - cosec x/h

Since cosec x = 1/sin x, therefore,

f'(x) = lim 1/sin (x+h) - 1/sin x/h

More simplification,

f'(x) = lim sin x - sin (x+h)/hsin x.sin (x+h)


f'(x) = lim sin x -sin (x+h)/h * lim 1/sin x.sin (x+h)

When h approaches to zero,

f'(x) = - cos x.1/sin2x

We can write it as;

f'(x) = -cot x.cosec x

Derivative of csc x by chain rule

To calculate the csc 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. The chain rule of derivatives is defined as;

dy/dx = dy/du x 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 cscx derivative by chain rule

To prove the csc derivative by using the chain rule, assume that,

f(x) = csc x = 1/sin x = (sin x)-1

By using chain rule of differentiation,

f'(x) = (-1) (sin x)-2 d/dx (sin x)


f'(x) = (-1/sin2x) . (cos x)


f'(x) = -cos x/ sin2x

Since sin x / cos x =tan x and 1/cos x = sec x, therefore we have proved the derivative of csc(x) as,

f'(x) = sec2 x . tan x

You can also use composite funciton calculator.

Derivative cscx using quotient rule

Another method for finding the csc x derivative is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated using the quotient rule. The quotient rule is defined as:

d/dx (f/g) = f(x). g'(x) -g(x).f'(x) /{g(x)}2

Proof of derivative of csc x by quotient rule

To prove the derivative of cscx, we can write it,

f(x) = csc x=1/ sin x =u/v

Supposing that u = 1 and v = sin x. Now by quotient rule,

f'(x) = (vu' - uv')/v2

f'(x) = [sin x d/dx(1) - 1. d/dx(sin x)] / (sin x)2

= [sin x (0) - 1 (cos x)] / sin2x

= (-cos x ) / sin2x

Hence, the derivatie of csc x is,

= -cot x . cosec x

You can also use quotient of two function calculator.

How to find the derivative cscx with a calculator?

The easiest way to calculate the csc derivative is by using an online tool. You can use our derivative derivative finder for this. Here, we provide you a step-by-step way to calculate derivative of cosecant x by using this tool.

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

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

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

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

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

Frequently Asked Questions

What is derivative from first principle?

The derivative from first principle is the measure of rate of change in a function. It is calculated by using the basic definition of derivative. It is also known as delta method.

What is cosec x equivalent to?

The cosecant function is a trigonometric function which is equivalent to the reciprocal of sine function. In triangle, it is the inverse of opposite side to the hypotenuse. It is written as;

Cosec x = hypotenuse/ opposite side

Is cos and cosec same?

No, cosine and cosecant are not same. The cosine function is the ratio of adjacent side to hypotenuse of a triangle. But the cosecant function is the ratio of opposite to the hypotenuse. Therefore, they are not same but can be parameters of same triangle.

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