Question

Let \[ f(x)=\sqrt{x^2+1}, \quad g(x)=\frac{x+1}{x^2+1}, \quad h(x)=2x-3 \] Then, \[ f'(h'(g'(x))) = ? \]

• (A) \( \frac{2}{\sqrt5} \)
• (B) \( \frac{-2}{\sqrt5} \)
• (C) \( 0 \)
• (D) \( \frac{1}{\sqrt5} \)

Hint:

Whenever an intermediate derivative is constant, substitute it immediately. It greatly reduces the complexity of nested functional problems.

Answer 1

The Correct Option is A

Concept:
In nested derivative problems, simplify the innermost derivative first. If an intermediate derivative becomes constant, the remaining expression becomes much easier.

Step 1: Finding \( h'(x) \).
Given, \[ h(x)=2x-3. \] Differentiating: \[ h'(x)=2. \] Since this derivative is constant, \[ h'(g'(x))=2. \]

Step 2: Finding \( f'(x) \).
Given, \[ f(x)=\sqrt{x^2+1}. \] Rewrite: \[ f(x)=(x^2+1)^{1/2}. \] Differentiating using the chain rule: \[ f'(x)=\frac12(x^2+1)^{-1/2}(2x). \] Simplifying: \[ f'(x)=\frac{x}{\sqrt{x^2+1}}. \]

Step 3: Evaluating the final expression.
We need: \[ f'(h'(g'(x))). \] Since \[ h'(g'(x))=2, \] therefore: \[ f'(2)=\frac{2}{\sqrt{2^2+1}}. \] Thus, \[ f'(2)=\frac{2}{\sqrt5}. \] Hence, \[ \boxed{\frac{2}{\sqrt5}}. \]