# Derivative of sec inverse x

Learn what is the derivative of sine inverse x with formula. Also understand how to prove the derivative of sine by first principle and inverse function formula.

Alan Walker-

Published on 2023-05-26

## Introduction derivative of sec inverse x

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

## What is the differentiation of sec inverse x?

The sec inverse x derivative with respect to the variable 'x' is expressed as d/dx(sec^-1(x)) = 1/|x|√x^2-1. This formula calculates the rate of change of the inverse trigonometric function sec x, where sec^-1(x) is the inverse of sec x. In a right triangle, secant is defined as the ratio of the hypotenuse to the adjacent side, or equivalently, as the reciprocal of cosine:

$\sec x=\frac{1}{\cos x}$

The derivative of sec^-1(x) is useful in various applications of calculus, including finding maximum and minimum values and solving optimization problems.

## Derivative of sec-1 x formula

The formula for the derivative of the inverse secant function is d/dx(sec^-1(x)) = 1/|x|√x^2-1, which can be written as;

$\frac{d}{dx}[\sec^{-1}x]=\frac{1}{|x|\sqrt{x^2-1}}$

This formula helps calculate the rate of change of the inverse trigonometric function sec x. It is important to note that the domain of sec^-1(x) is limited to $(-\infty, -1]$ and $[1, \infty)$, which means the derivative is only defined within this range.

## How do you prove the derivative of sec-1x?

There are multiple ways to derive the derivative of sec inverse. Some of these derivative rules are;

1. First Principle
2. Inverse function formula

Each method provides a different way to compute the inverse secant derivative. By using these methods, we can mathematically prove the formula for finding the arcsec derivative.

## Derivative of sec inverse x by first principle

According to the first principle of derivative, the differentiation of sec inverse x is equal to 1/|x|√x^2-1. The derivative of a function by the 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 allows us to determine the rate of change of a function at a specific point by using the limit definition of the derivative. You can also use our derivative by definition calculator as it also follows the above formula.

## Proof of derivative of secant inverse by first principle

To prove the derivative of inverse sec x by using first principle, we start by replacing f(x) by sec-1x.

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

So,

$f'(x)=\lim_{h\to 0} \frac{\sec^{-1}(x+h)-\sec^{-1}x}{h}$

We will use some inverse trigonometric identities and formulas to verify the derivative of sec inverse. These are:

$\tan^{-1}a-\tan^{-1}b=\tan^{-1}\left(\frac{a-b}{1+ab}\right)$

$\sec^{-1}x=\frac{1}{\cos^{-1}}$

Suppose that,

$\sec y=x$

And,

$\sec^2y=1+\tan^2y$

Or,

$\tan^2y=\sec^2y-1$

Taking square root on both sides,

$\tan y=\sqrt{\sec^2y-1}$

Or,

$\tan y=\sqrt{x^2-1}$

$y=\tan^{-1}\sqrt{x^2-1}$

Now use this substitution in the above derivative formula, Now by using tan inverse formula, Now simplifying by multiplying and dividing with,

$\sqrt{(x+h)^2-1}+\sqrt{x^2-1}$ Then, As h approaches to zero, As the above calculations show that the derivative of sec inverse x is not easily solvable by hand. You can use our derivative by definition calculator to solve these calculations within a few seconds.

## Arcsec derivative using inverse function formula

The inverse function formula is a fundamental technique for finding the derivatives of inverse functions. Mathematically this formula is expressed as:

$[f^{-1}]'(x)=\frac{1}{f'[f^{-1}(x)]}$

This formula is useful to calculate the rate of change of inverse trigonometric functions, such as sec^-1(x). By using the inverse function formula, we can derive the sec inverse x derivative in terms of the derivative of sec x.

## Proof of derivative of arcsec by inverse function formula

To prove differentiation of sec inverse x, we can start by assuming that,

$y=\sec^{-1}x$

Then, we can write the above equation as;

$\sec y=x$

Since, differentiating an equation of two independent variables is known as implicit differentiation, therefore from above equation,

$\tan y\sec y\frac{dy}{dx}=1$

Where the derivative of sec x is sec xtan x. Now rearranging the above equation,

$\frac{dy}{dx}=\frac{1}{\tan y\sec y}$

Since y = sec-1x.

$\frac{dy}{dx}=\frac{1}{\tan(\sec^{-1}x)\sec(\sec^{-1}x)}$

Assume that sec-1x = θ then sec θ = x, and since we know that,

$\sec^2\theta=\tan^2\theta+1$

Or,

$\tan^2\theta=\sec^2\theta-1=x^2-1$

Taking Square root,

$\tan \theta=\sqrt{x^2-1}$

Similarly,

$\sec(\sec^{-1}x)=\sec \theta=x$

Substituting these values in the derivative formula,

$\frac{dy}{dx}=\frac{1}{x\sqrt{x^2-1}}$

Hence we have proved the sec inverse derivative using the inverse function theorem. This theorem is suitable for inverse trig function differentiation.

## How to find the derivative of arcsec x with a calculator?

The easiest way to calculate the derivative of secant inverse is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as sec^-1x 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 sec inverse x.
2. Now, select the variable by which you want to differentiate sec-1x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate secant inverse 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, the inverse function derivative calculator will provide you the derivative of secant inverse x within a few seconds.