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).