How to Find the Derivative of Arcsin(x) Using the Chain Rule

Derivative of arcsin(x)

The derivative of arcsin(x), denoted as d/dx arcsin(x) or simply (arcsin(x))’, can be found using the chain rule of differentiation

The derivative of arcsin(x), denoted as d/dx arcsin(x) or simply (arcsin(x))’, can be found using the chain rule of differentiation.

To differentiate arcsin(x), we start by recalling the defining relationship of arcsin(x):

sin(arcsin(x)) = x

Now, we can differentiate both sides of this equation with respect to x:

d/dx (sin(arcsin(x))) = d/dx (x)

Using the chain rule, the derivative of sin(arcsin(x)) can be obtained as follows:

cos(arcsin(x)) * d/dx (arcsin(x)) = 1

Simplifying this equation, we have:

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

To find the value of cos(arcsin(x)), we can utilize the Pythagorean identity:

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

Since sin(arcsin(x)) = x, we can substitute this into the equation:

x^2 + cos^2(arcsin(x)) = 1

Now, solving for cos(arcsin(x)), we get:

cos(arcsin(x)) = sqrt(1 – x^2)

Finally, substituting this value back into the derivative equation, we obtain:

d/dx (arcsin(x)) = 1 / sqrt(1 – x^2)

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

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