The Derivative of f(cos⁻¹x) Using the Chain Rule

f'(cos⁻¹x)

To find the derivative of the function f(cos⁻¹x), we can use the chain rule

To find the derivative of the function f(cos⁻¹x), we can use the chain rule. The chain rule states that if we have a composite function, such as f(g(x)), the derivative of this composition is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

In this case, the outer function is f(u) and the inner function is g(x) = cos⁻¹x.

First, let’s find the derivative of the outer function f(u). Without knowing the specific function f(u), we cannot determine its derivative. Therefore, we will leave it as f'(u) for now.

Next, let’s find the derivative of the inner function g(x) = cos⁻¹x. This can be found using the inverse trigonometric identities. The derivative of cos⁻¹x with respect to x is equal to -1 / √(1 – x²). Therefore, g'(x) = -1 / √(1 – x²).

Now, we can apply the chain rule. The derivative of f(cos⁻¹x) with respect to x, denoted as (f(cos⁻¹x))’, is equal to f'(u) * g'(x). Since u = cos⁻¹x, we substitute f'(u) with f'(cos⁻¹x) and g'(x) with -1 / √(1 – x²) in the chain rule formula.

Therefore, (f(cos⁻¹x))’ = f’ (cos⁻¹x) * (-1 / √(1 – x²)).

That is the derivative of f(cos⁻¹x) with respect to x.

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