Derivative of arcsin(x): Applying the chain rule to differentiate f(x) = arcsin(x) with respect to x.

d/dx arcsin(x)

To differentiate the function f(x) = arcsin(x) with respect to x, we can use the chain rule

To differentiate the function f(x) = arcsin(x) with respect to x, we can use the chain rule. The chain rule states that if we have a composition of functions, f(g(x)), then the derivative of that composition is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

First, let’s rewrite the function arcsin(x) as f(x) = sin^(-1)(x).

Now, we can differentiate f(x) using the chain rule. Let u = x and g(u) = sin(u). So, f(x) = g^(-1)(u).

Now, we differentiate g(u) = sin(u) with respect to u:

d/du [sin(u)] = cos(u).

Next, we differentiate f(x) = g^(-1)(u) with respect to x:

d/dx [f(x)] = d/dx [g^(-1)(u)].

Applying the chain rule, the derivative of f(x) with respect to x is:

d/dx [f(x)] = d/du [g^(-1)(u)] * d/dx [u].

We know that d/du [g^(-1)(u)] is equal to (1 / d/dx [g(u)]) evaluated at u = x.

So, d/dx [f(x)] = (1 / d/dx [g(x)]) * d/dx [u].

Substituting g(x) = sin(x) and d/dx [u] = 1, we have:

d/dx [f(x)] = (1 / cos(x)) * 1.

Therefore,

d/dx [arcsin(x)] = 1 / cos(x).

In summary, the derivative of the arcsin(x) with respect to x is equal to 1 / cos(x).

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