Discover The Derivative Of Sin^-1(X) – Simplified Explanation And Formula

d/dx[sin^-1(x)]=

1/sqrt(1-x^2)

To find the derivative of sin^-1(x), we first need to use the inverse property of trigonometric functions and express sin^-1(x) in terms of the inverse sine function:

sin(sin^-1(x)) = x

Now, we take the derivative of both sides of this equation with respect to x:

cos(sin^-1(x)) * d/dx[sin^-1(x)] = 1

Here, we have used the chain rule on the left-hand side of the equation. Now, we can solve for d/dx[sin^-1(x)]:

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

To simplify this expression further, we use the Pythagorean identity: sin^2(x) + cos^2(x) = 1. Since sin^-1(x) represents an angle whose sine is x, we can write:

sin^-1(x) + cos^2(sin^-1(x)) = 1

Solving for cos(sin^-1(x)), we get:

cos(sin^-1(x)) = sqrt(1 – sin^2(sin^-1(x))) = sqrt(1 – x^2)

Substituting this into our expression for the derivative, we get:

d/dx[sin^-1(x)] = 1 / sqrt(1 – x^2)

Therefore, the derivative of sin^-1(x) is 1 / sqrt(1 – x^2).

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