Suppose f^-1 is the inverse function of a differentiable function f and let G(x)=1 / f^-1(x). If

step 1 In this problem, we're given that the functions F and G are inverse functions, so F of X is equa

Suppose f^-1 is the inverse function of a differentiable...

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Key Concepts

Inverse Functions

An inverse function reverses the operation of the original function, such that if f(a) = b then f?¹(b) = a. This relationship is central in problems where one needs to move between the input and output of a function, and it underpins many areas of calculus and analysis.

Derivative of Inverse Functions

The derivative of an inverse function is given by the formula (f?¹)'(y) = 1 / f'(f?¹(y)). This formula is essential when calculating derivatives of compositions involving inverse functions, as it links the behavior of a function to the behavior of its inverse.

Differentiation of Reciprocal Functions

When differentiating a function that is defined as the reciprocal of another function, such as G(x) = 1 / h(x), the derivative can be found using the rule G'(x) = -h'(x) / [h(x)]². This technique is a straightforward application of the chain rule combined with the power rule.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This rule is crucial in differentiating functions defined in terms of other functions, including inverse functions.

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