The derivative of a function is a fundamental concept in calculus, providing insight into the rate of change of that function at a specific point. In this essay, we will focus on the derivative of the arcsine function, denoted as y = arcsin(x). We will explore the formula for finding this derivative, present a rigorous proof of the formula using implicit differentiation, and provide examples to illustrate its application.

## The formula for the Derivative of arcsin(x)

The derivative of the arcsine function, often denoted as dy/dx, can be expressed as:

dy/dx = 1 / √(1 – x^2)

This formula provides the rate of change of the arcsine function with respect to the input variable x. To better understand how this formula is derived, we will now present a rigorous proof using implicit differentiation.

## Proof of the Derivative of arcsin(x)

- Begin with the arcsine function: y = arcsin(x).
- Express this relationship in trigonometric terms: sin(y) = x.
- Apply implicit differentiation to both sides of the equation with respect to x:d/dx[sin(y)] = d/dx[x]
- By using the chain rule, the left side becomes:cos(y) * dy/dx = 1
- Solve for dy/dx:dy/dx = 1 / cos(y)
- To find cos(y), we can use the Pythagorean identity for sin(y):sin^2(y) + cos^2(y) = 1
Rearrange to solve for cos^2(y):

cos^2(y) = 1 – sin^2(y)

- Substituting sin^2(y) with x^2 (from step 2), we get:cos^2(y) = 1 – x^2
- Finally, to find cos(y), take the square root:cos(y) = √(1 – x^2)

The proof establishes that the derivative of the arcsine function is dy/dx = 1 / cos(y), where cos(y) is given by cos(y) = √(1 – x^2).

Examples of Finding the Derivative of arcsin(x)

**Example 1:** Let’s find the derivative of y = arcsin(x).

dy/dx = 1 / √(1 – x^2)

If x = 0.5, then:

dy/dx = 1 / √(1 – (0.5)^2) = 1 / √(1 – 0.25) = 1 / √0.75 ≈ 1.155

So, the derivative of arcsin(0.5) is approximately 1.155. This represents the rate of change of the arcsine function at x = 0.5.

**Example 2:** Consider the function y = arcsin(0.8). To find its derivative, we can use the formula:

dy/dx = 1 / √(1 – x^2)

Substitute x = 0.8 into the formula:

dy/dx = 1 / √(1 – (0.8)^2) = 1 / √(1 – 0.64) = 1 / √0.36 = 1 / 0.6 = 1.67

The derivative of y = arcsin(0.8) is 1.67.

The derivative of the arcsine function, dy/dx = 1 / √(1 – x^2), is a powerful tool in calculus for understanding the rate of change of the arcsine function at a given point. Through rigorous proof using implicit differentiation, we have demonstrated the validity of this formula. Additionally, examples have shown how to apply the formula to find the derivative of specific values of the arcsine function. Understanding the derivative of the arcsine function is valuable in various mathematical and scientific applications, where the rate of change of angles and trigonometric functions plays a significant role.