How to Find the Derivative of the Arcsin(x) Function Using the Chain Rule and Pythagorean Identity

Derivative of arcsin(x)

To find the derivative of the arcsin(x) function, we can use the chain rule

To find the derivative of the arcsin(x) function, we can use the chain rule. Let’s start from the definition of the arcsin(x) function:

y = arcsin(x)

Taking the sin of both sides, we have:

sin(y) = x

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

cos(y) * dy/dx = 1

Now, we need to solve for dy/dx, which is the derivative of y with respect to x:

dy/dx = 1 / cos(y)

To find cos(y) in terms of x, we can use the Pythagorean identity sin^2(y) + cos^2(y) = 1. Since sin(y) = x, we can substitute:

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

cos^2(y) = 1 – x^2

cos(y) = √(1 – x^2)

Substituting this back into the equation for dy/dx, we have:

dy/dx = 1 / √(1 – x^2)

So, the derivative of arcsin(x) is:

dy/dx = 1 / √(1 – x^2)

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