Finding the Derivative of the Inverse Sine Function | Step-by-Step Guide and Formula

derivative of sin^-1

The derivative of the inverse sine function, denoted as sin^(-1)(x) or arcsin(x), can be found using differentiation rules

The derivative of the inverse sine function, denoted as sin^(-1)(x) or arcsin(x), can be found using differentiation rules. Let’s go through the steps to find the derivative.

We start with the formula for the inverse sine function:
y = sin^(-1)(x)

To find the derivative, we need to express y in terms of x and differentiate with respect to x. However, it is challenging to express y solely in terms of x. So, we can use the fact that sin(x) and arcsin(x) are inverse functions to help us simplify the expression.

The inverse function property states that if y = sin^(-1)(x), then x = sin(y).

Let’s rewrite the equation in terms of x and differentiate implicitly.

x = sin(y)

Now, we differentiate both sides of the equation with respect to x:

d/dx (x) = d/dx (sin(y))

The left-hand side is simply 1 (since x doesn’t depend on x itself), and the right-hand side requires the chain rule since y is a function of x.

1 = cos(y) * dy/dx

To simplify further, we need to express cos(y) in terms of x. Using the Pythagorean identity sin^2(y) + cos^2(y) = 1, we know that cos(y) = sqrt(1 – sin^2(y)).

Since x = sin(y), we can substitute x into the expression for sin(y) to get cos(y) = sqrt(1 – x^2).

Now, our equation looks like:

1 = sqrt(1 – x^2) * dy/dx

To solve for dy/dx, we isolate it:

dy/dx = 1 / sqrt(1 – x^2)

And there you have it! The derivative of the inverse sine function, dy/dx, is equal to 1 divided by the square root of (1 – x^2).

More Answers: