What is the derivative of sin^-1(x)?

Solution:

A derivative is the output resulting from differentiating a function with respect to its variable.

Let us assume,

y = sin^-1(x)   .........(i)

⇒ sin(y) = x

Differentiating with respect to y, we get

dx/dy = cos(y)  

Thus,

dy/dx = 1/cos(y)    .........(ii)

Substituting (i) in (ii),

We get, dy/dx = 1/cos(sin^-1(x))

Using the trignometric identity below and equation (i),

cos²(y) = 1 - sin²(y) 

= 1 - (sin(sin^-1(x)))² 

= 1 - x² 

Thus, cos²(y) = 1 - x² 

Taking square root on both the sides,

cos(y) = √(1 - x²)

Plugging this into equation (ii), we finally get

dy/dx = 1/√(1 - x²), -1 < x < 1

Thus, the derivative of  sin^-1x = dy/dx is 1/√(1 - x²), -1 < x < 1

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What is the derivative of sin^-1(x)?

Summary:

The derivative of sin^-1x = dy/dx is 1/√(1 - x²), -1 < x < 1