Derivative of Arcsin

Learn about the derivative of sine inverse x with formula. Also find how to prove the derivative of arcsin by first principle and implicit differentiation.

Alan Walker-

Published on 2023-05-26

Introduction to the derivative of arcsin

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 arcsin derivative by applying the first principle of differentiation. In this article, you will learn what the derivative of sine inverse x is and how to calculate the derivative of sine inverse by using different approaches.

What is the derivative of sin-1 x?

The derivative of the arcsine function, also known as the inverse sine function, can be found by using the formula d/dx(sin^-1x) = 1/√(1-x^2). This formula is derived using the first principle method and can be used to find the rate of change of the arcsine function with respect to the variable x. The arcsine function is the inverse of the ratio of opposite to hypotenuse in a right triangle, and it is denoted by sin^-1x or arcsin(x).

Derivative arcsin formula

The derivative of the arcsine function, also known as the inverse sine function, is given by the formula

d / dx(sin-1x) = 1/√1-x2

This formula can be derived using the first principle method and is closely related to the derivative of cos inverse function. In fact, the formula for the derivative of arcsine can be written as the negative of the derivative of cosine inverse, i.e., d/dx(cos^-1x) = -1/√(1-x^2). The arcsine function is denoted by sin^-1x or arcsin(x) and is the inverse of the ratio of opposite to hypotenuse in a right triangle.

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

There are various ways to prove the arcsin derivative. These are:

1. First Principle

2. Implicit differentiation

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 inverse sin x.

Differentiation of sin inverse x by first principle

The first principle method for finding the derivative of the arcsine function, sin^-1x, involves using algebra to find a general expression for the slope of a curve. This method is also known as the delta method and is used to find the instantaneous rate of change of a function. The derivative of a function at a point is defined as the limit of the difference quotient as the change in the input variable approaches zero. Mathematically, the derivative of a function f(x) with respect to x is given by the limit definition of derivative calculator;

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

Proof of derivative of sine inverse by first principle

To prove the arcsin differentiation by using first principle, replace f(x) by sin x.

f'(x)=limh→0f(x+h)-f(x)/h

So,

f'(x)=limh→0sin-1 (x+h)-sin-1 x /h

Suppose that,

sin-1 (x+h) = A and sin-1 x = B

Also, h = x+h-h = sin B - sin A therefore, as h approaches zero, A will approach B.

f'(x)=limA→B A-B /sin A - sin B

Using the formula sin A - sin B = 2cos(A+B)/2sin(A-B)/2

f'(x)=limA→B A-B /2cos(A+B)/2sin(A-B)/2

Or,

f'(x)=limA→B (A-B)/2 /cos(A+B)/2sin(A-B)/2

Let A-B/2 = t then

f'(x)=limA→B 1 /cos(A+B)/2 * )=limt→0 t/sin t

As A approaches B and t approaches zero,

f'(x) = 1/cos B = 1/√1-x2

Arcsin differentiation using implicit differentiation

implicit differentiation is a technique used to find the derivative of a function that is defined implicitly by an equation involving two or more variables. We can use this method to prove the differentiation of sin inverse x. .

Proof of derivative of arcsin by implicit differentiation

To prove derivative of inverse sine, let us assume,

y = sin-1x.

Then, we can write the above equation as;

sin y= x

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

(cos y) dy/dx = 1

By using trigonometric identities,

sin2y + cos2y = 1

→ cos2y + x2 = 1

→ cos2y = 1 - x2

Taking square root on both sides,

→cos y = √(1 - x2)

Substituting the above value in (i), we get

√(1 - x2) dy/dx = 1

By rearranging we get,

dy/dx = 1/√(1 - x2),

Hence we have proved the derivative sine inverse using implicit differentiation. Also calculate the derivative of sec inverse x by using implicit differentiation.

How to find the arcsin derivative with a calculator?

The easiest way to calculate the derivative of sine inverse is by using an online differentiation 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 sin-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 sin-1x.

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

3. Select how many times you want to differentiate sine 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 derivative of sine inverse x within a few seconds.

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

What is the first principle of derivation?

The derivative is the measure of rate of change in a function. It can be the rate of change in distance with time or rate of change in temperature with respect to distance. The first principle of derivation says the rate of change in a function with respect to the independent variable involved.

What is the derivative of sin inverse x?

The derivative of an inverse function sine is equal to the negative derivative of the cosine function. It is written as;

d/dx(sin-1x) = 1/√1-x2

What is meant by sine function?

The sine inverse function is the inverse ratio of opposite to hypotenuse of a triangle. It is written as sin-1x. But sometimes, it is also written as arcsin x. In mathematical form, it is written as;

x=sin-1(opposite/hypotenuse)