Learn How to Find the Derivative of Arcsin(x) with this Step-by-Step Guide | Math Tutorial

Derivative of arcsin(x)

du/sqrt(1-u^2)

To find the derivative of arcsin(x), we can use the formula for the derivative of an inverse function, which says:

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

Let’s use this formula with f(x) = sin(x) and f^-1(x) = arcsin(x):

(d/dx) [arcsin(x)] = 1 / [sin'(arcsin(x))]

Now, we need to find the derivative of sin(x) and substitute the inverse function arcsin(x). The derivative of sin(x) is cos(x), so:

(d/dx) [arcsin(x)] = 1 / [cos(arcsin(x))]

To simplify further, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find cos(x) in terms of sin(x):

cos(x) = sqrt(1 – sin^2(x))

Substituting sin(x) with x and arcsin(x) with θ, we get:

cos(θ) = sqrt(1 – x^2)

Therefore, the derivative of arcsin(x) is:

(d/dx) [arcsin(x)] = 1 / [sqrt(1 – x^2)]

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