Introduction to the derivative of cos inverse

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of inverse cosine can be calculated by following the rules of differentiation. Or, we can directly find the cos-1x derivative by applying the first principle of differentiation. In this article, you will learn what the cos inverse x differentiation is and how to calculate the differentiation of cos inverse by using different approaches.

What is the derivative of cos-1 x?

The derivative of cos inverse x with respect to the variable x is equal to $\frac{-1}{\sqrt{1-x^2}}$. This is commonly denoted by d/dx(cos-1x). It is the inverse of the rate of change of the trigonometric function cos x. The inverse cosine function represents the angle whose cosine equals a given value x and is written as:

$\x \;=\; \cos^{-1}\frac{adjacent \; side}{hypotenuse}$

In other words, the inverse cosine function can be used to find the angle of a right triangle if the lengths of the adjacent and hypotenuse sides are known.

Cos inverse x derivative formula

The formula for finding the derivative of the inverse cosine function is equal to the negative of the sine inverse derivative, that is;

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

This formula is the negative of the sine inverse derivative. It tells us how the rate of change of the inverse cosine function varies with respect to its input variable. Specifically, it gives us the instantaneous rate of change at any given point on the function's curve. The denominator of the formula, the square root of one minus x squared, ensures that the function remains continuous and differentiable within its domain. Knowing this formula is essential for solving problems in calculus and related fields.

How do you prove the derivative of cos-1x?

There are different derivative rules to derive cos inverse derivative. These methods allow you to find the rate of change or slope of the cos^-1 x function at any point, which is useful in many applications. We can prove the differentiation of cos^-1 by using;

1. First Principle

2. Implicit differentiation

Let's prove the inverse cosine derivative by using first principle.

Derivative of cos inverse by first principle

The derivative first principle tells that the cos inverse derivative is equal to the negative of derivative of sine inverse of 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 calculator formula is represented as:

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

Proof of derivative of cosine inverse by first principle

To prove the derivative of cos inverse x by using the first principle, we start by replacing f(x) by cos inverse x. Or, replace f(x) by cot x to find the differential of cot x.

$f(x) \;=\; \lim_{h\to0} f(x+h)\;-\;\frac{f(x)}{h}$

Since by trigonometric inverse formulas, we know that,

$\cos^{-1}x \;+\; \sin^{-1}x \;=\; \frac{π}{2}$

Therefore, we will find the derivative of sine inverse to calculate derivative of inverse of cosine.

$\frac{d(\sin^{-1}x)}{dx} \;=\; \lim_{h\to0}\sin^{-1}(x+h) \;-\; \sin^{-1} (x + h) \;=\; \frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}(\cos^{-1}x) \;=\; \lim_{h\to0} \cos^{-1}(x+h) - \cos^{-1} \frac{x}{h}$

Simplifying more leads us to the cos inverse x differentiation,

$=\lim_{h\to0} \frac{π}{2}-\sin^{-1}(x+h)-\frac{π}{2}+\sin^{-1} \frac{x}{h}$

$=\lim_{h\to0}-\sin^{-1}(x+h) \;+\; \sin^{-1} \frac{x}{h}$

$=\lim_{h\to0} \sin^{-1} 1(x+h)- \sin^{-1} \frac{x}{h} \;=\;\frac{-d(\sin^{-1}x)}{dx}$

Hence the differentiation of cos inverse x is,

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

This method is also applicable to calculate the derivative of sec inverse x.

Derivative of cos inverse x using implicit differentiation

Since in implicit differentiation, we differentiate a function with two variables. Here we will prove the derivative of inverse cos x by implicit differentiation.

Proof of derivative of cos inverse by implicit differentiation

To prove differentiation of cos^-1x by inverse derivative formula, let us assume,

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

Then, we can write the above equation as;

$\cos y = x$

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

$(-\sin y) \frac{dy}{dx} \;=\; 1$

By using trigonometric identities,

$\sin^2y + \cos^2y \;=\; 1$

$\sin^2y + x^2 \;=\; 1$

$\sin^2y \;=\; 1 - x^2$

Taking square root on both sides,

$\sin y \;=\; \sqrt{(1 - x^2)}$

Substituting the above value in (i), we get

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

By rearranging we can get the inverse cosine derivative,

$\frac{dy}{dx} \;=\; \frac{-1}{\sqrt{(1 - x^2)}},$

Hence we have proved the cos^-1 derivative using implicit differentiation. This method is applicable for those functions which are implicit in nature. You can also use implicit differentiation calculator which will guide you in calculations.

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

The easiest way to calculate the derivative of cosine inverse is by using an online tool. You can use our differentiation calculator for this. Here, we provide you a step-by-step way to calculate cos^-1(x) derivative by using this tool.

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

2. Now, select the variable by which you want to differentiate cos^-1x. Here you have to choose x.

3. Select how many times you want to differentiate cosine 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, you will get the differentiation of cos inverse x within a few seconds.

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

Frequently Asked Questions

What's the derivative of inverse sine?

The derivative of arcsin is written as d(sin^-1x) / dx = 1 √1-x². It is the rate of change of inverse sine function. It is written as;

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

How to find derivative of a function?

The derivative of a function can be easily calculated by using following steps;

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

1. For f(x) find f(x+h).

2. Use the above values in first principle of derivative as;

3. Simplify the equation and take the limit as h approaches to zero to get derivative of a given function.

What is cos inverse x formula?

The cosine inverse function is the inverse ratio of adjacent to hypotenuse of a triangle. It is written as cos-1x. But sometimes, it is also written as arccos x. In mathematical form, it is written as;

$x = \cos^{-1}\left(adjacent side / hypotenuse\right)$