The derivative of the arcsin function (also denoted as sin^(-1)(x) or asin(x)) can be found using differentiation techniques. Let’s denote y = arcsin(x), where -1 ≤ x ≤ 1.

To find the derivative dy/dx, we can differentiate both sides of the equation y = arcsin(x) with respect to x. Using the chain rule, we have:

dy/dx = d/dx(arcsin(x))

Now, let’s differentiate the right side of the equation. The derivative of arcsin(x) can be found as follows:

Let y = arcsin(x) Then, x = sin(y)

Differentiating both sides with respect to x:

1 = cos(y) * dy/dx

Now, we can solve for dy/dx:

dy/dx = 1 / cos(y)

To determine the value of cos(y), we can use the trigonometric identity:

sin^2(y) + cos^2(y) = 1

Since x = sin(y), we can substitute sin^2(y) = x^2:

x^2 + cos^2(y) = 1

Solving for cos(y):

cos(y) = sqrt(1 – x^2)

Substituting this value into the expression for dy/dx:

dy/dx = 1 / sqrt(1 – x^2)

Therefore, the derivative of the arcsin function is given by:

dy/dx = 1 / sqrt(1 – x^2)

It’s worth noting that this derivative is valid within the domain -1 ≤ x ≤ 1, as arcsin(x) is only defined within that range.

Here are three examples demonstrating the application of the derivative of the arcsin function:

Example 1: Let’s find the derivative of y = arcsin(2x).

Using the derivative formula derived earlier, we have:

dy/dx = 1 / sqrt(1 – (2x)^2)

Simplifying:

dy/dx = 1 / sqrt(1 – 4x^2)

So, the derivative of y = arcsin(2x) is dy/dx = 1 / sqrt(1 – 4x^2).

Example 2: Consider the function y = 3arcsin(5x).

Using the derivative formula, we have:

dy/dx = 1 / sqrt(1 – (5x)^2)

Simplifying:

dy/dx = 1 / sqrt(1 – 25x^2)

Thus, the derivative of y = 3arcsin(5x) is dy/dx = 1 / sqrt(1 – 25x^2).

Example 3: Let’s find the derivative of y = -arcsin(3x^2).

Using the derivative formula, we have:

dy/dx = 1 / sqrt(1 – (3x^2)^2)

Simplifying:

dy/dx = 1 / sqrt(1 – 9x^4)

Therefore, the derivative of y = -arcsin(3x^2) is dy/dx = 1 / sqrt(1 – 9x^4).

These examples illustrate how to find the derivatives of functions involving the arcsin function using the derived formula dy/dx = 1 / sqrt(1 – x^2).