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.