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: Understanding the Question:
Work from the inside. We need \(g'(x)\), then \(h'\) of it, then \(f'\) of that.

Step 2: Inner derivatives:
\(h(x) = 2x - 3\), so \(h'(x) = 2\) for every input. Hence \(h'(g'(x)) = 2\), no matter what \(g'(x)\) is.
(For reference, \(g'(x) = \frac{1-2x-x^2}{(x^2+1)^2}\), but its value is not needed.)

Step 3: Outer derivative:
\(f(x) = \sqrt{x^2+1}\), so \(f'(x) = \frac{x}{\sqrt{x^2+1}}\).
\[ f'(2) = \frac{2}{\sqrt5} \]

Step 4: Other options:
Options B and D still contain \(x\), but the expression has no \(x\) left. Option A would need \(f'(0)\).

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