How To Calculate The Derivative Of The Arcsine Function Using The Formula

derivative of sin^-1

1/√(1-x^2)

The derivative of arcsine (sin^-1) function can be calculated using the formula:

d/dx (sin^-1 x) = 1 / √(1 – x^2)

This means that the derivative of the arcsine function is equal to the reciprocal of the square root of one minus x squared.

To understand how to derive this formula, we start with the inverse trigonometric identity:

sin(sin^-1 x) = x

Differentiating both sides of the equation with respect to x gives:

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

Solving for d/dx (sin^-1 x), we get:

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

Using the Pythagorean identity, we have:

1 – sin^2(sin^-1 x) = cos^2(sin^-1 x)

Substituting this into the previous equation gives:

d/dx (sin^-1 x) = 1 / √(1 – sin^2(sin^-1 x))

Using the definition of the arcsine function, we have:

sin(sin^-1 x) = x

Therefore:

d/dx (sin^-1 x) = 1 / √(1 – x^2)

This is the final formula for the derivative of the arcsine function.

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