The Derivative of the Inverse Sine Function: How to Find it Using the Chain Rule and Trigonometric Identities

(d/dx) sin^-1x

To find the derivative of the inverse sine function, we can use the chain rule

To find the derivative of the inverse sine function, we can use the chain rule. Recall that the chain rule states that if we have a composition of functions, the derivative of the composition is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Let’s proceed with the problem:

First, let’s define y = sin^(-1)(x). This means that y is the angle whose sine is x.

Next, we can rewrite this equation using inverse trigonometric function notation:

sin(y) = x.

Now, let’s take the derivative of both sides of the equation with respect to x:

d/dx [sin(y)] = d/dx [x].

Using the chain rule, the left-hand side becomes:

cos(y) * dy/dx.

Since sin(y) = x, we can rewrite the equation as:

cos(y) * dy/dx = 1.

Now, we can solve for dy/dx, which is the derivative we are looking for:

dy/dx = 1 / cos(y).

We can substitute y = sin^(-1)(x) back into the equation:

dy/dx = 1 / cos[sin^(-1)(x)].

Recall that cos[sin^(-1)(x)] represents the cosine of the angle whose sine is x. Using trigonometric identities, we know that cos[sin^(-1)(x)] = √(1 – x^2).

Therefore, the final derivative is:

dy/dx = 1 / √(1 – x^2).

So, the derivative of sin^(-1)(x) with respect to x is 1 / √(1 – x^2).

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