If f and g are differentiable functions such that g^'(a)=2 and g(a)=b and if f o g is an identit

step 1 In this problem we have G of x as f of 2 fx plus 2 whole square G dash of x is given as 2 times

If f and g are differentiable functions such that g^'(a)=2...

If $f$ and $g$ are differentiable functions such that $g^{\prime}(a)=2$ and $g(a)=b$ and if $f o g$ is an identity function, then $f^{\prime}(b)$ has the value equal to
(a) $2 / 3$
(b) 1
(c) 0
(d) $1 / 2$

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Key Concepts

Inverse Function Theorem

This theorem relates the derivative of an inverse function to the derivative of the original function. Specifically, if f and g are inverse functions (i.e., f(g(x)) = x for all x in the domain of g), then the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. This concept underpins the method for finding f'(b) when knowing information about g.

Chain Rule

The chain rule is a fundamental formula in calculus for differentiating composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) multiplied by g'(x). In problems involving f o g being an identity, the chain rule helps relate the various derivatives of f and g at specific points.

Differentiability and Identity Functions

A function is differentiable if it has a derivative at every point in its domain, meaning the function is locally linearizable. When a composite function, such as f o g, equals the identity function, its derivative must be 1 at every point. This scenario is used to establish specific relationships between the derivatives of the composing functions.

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