A Guide to Finding the Derivative of the Inverse Sine Function using Implicit Differentiation and the Chain Rule

d/dx arcsin(x)

To find the derivative of the inverse sine function, arcsin(x), with respect to x, we can use the chain rule

To find the derivative of the inverse sine function, arcsin(x), with respect to x, we can use the chain rule.

The inverse sine function can be denoted as y = arcsin(x). Taking the sine on both sides of the equation gives sin(y) = x. Now, we can find the derivative dy/dx using implicit differentiation.

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

cos(y) * dy/dx = 1

Next, we can solve for dy/dx by isolating it:

dy/dx = 1 / cos(y)

Now, recall the Pythagorean identity: sin^2(y) + cos^2(y) = 1. Since y = arcsin(x), we can rewrite sine in terms of x:

sin(y) = x

Using the Pythagorean identity, we have:

1 – cos^2(y) = x^2

cos^2(y) = 1 – x^2

cos(y) = sqrt(1 – x^2)

Substituting this into the equation dy/dx = 1 / cos(y), we get:

dy/dx = 1 / sqrt(1 – x^2)

Therefore, the derivative of arcsin(x) with respect to x is:

d/dx arcsin(x) = 1 / sqrt(1 – x^2)

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