Derivative of arcsin(x): Finding the derivative of the function f(x) = arcsin(x) using the chain rule

d/dx arcsin(x)

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

To find the derivative of the function f(x) = arcsin(x), we can use the chain rule. The chain rule states that if we have a function g(x) inside another function h(x), then the derivative of h(g(x)) with respect to x is equal to the derivative of h with respect to g multiplied by the derivative of g with respect to x.

In this case, g(x) = x and h(x) = arcsin(x). The derivative of h(g(x)) with respect to x is equal to the derivative of h with respect to g multiplied by the derivative of g with respect to x.

The derivative of h(x) = arcsin(x) with respect to g(x) can be found by taking the derivative of arcsin(x) with respect to x and then dividing it by the derivative of g(x) = x with respect to x.

The derivative of arcsin(x) with respect to x can be found using inverse trigonometric identities. Recall that the derivative of arcsin(x) with respect to x is equal to 1 / √(1 – x^2).

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

d/dx (arcsin(x)) = 1 / √(1 – x^2)

Now, dividing this by the derivative of g(x) = x with respect to x, which is 1, we get:

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

Simplifying this expression, we have:

d/dx (arcsin(x)) = 1 / √(1 – x^2)

So, the derivative of f(x) = arcsin(x) with respect to x is 1 divided by the square root of (1 – x^2).

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