Question

If \(f(x) = \sqrt{x^2+1},g(x) = \frac{x+1}{x^2+1},h(x) = 2x-3\), then \(f^'(h^'(g^'(x))) =\)

• A. \(0\)
• B. \(\frac{1}{\sqrt{x^2+1}}\)
• C. \(\frac{2}{\sqrt{5}}\)
• D. \(\frac{x}{\sqrt{x^2+1}}\)

Hint:

\(h'(x)\) is the constant 2, so \(f'(h'(g'(x))) = f'(2)\).

Answer 1

The Correct Option is C

Step 1: Plan:
Spot that the derivative of a linear function is constant, which collapses the composite.

Step 2: Steps:
Because $h$ is linear with slope $2$, the middle step $h'(g'(x))$ is always $2$. That is a number, so the answer cannot depend on $x$.
The outer step is $f'(2) = \frac{2}{\sqrt{4+1}} = \frac{2}{\sqrt5}$.

Final Answer:
The value is $\frac{2}{\sqrt5}$, option (C). \[ \boxed{\frac{2}{\sqrt5}} \]