Question

If \( f(x) = \int \cos^{-1} \left( \frac{x}{\sqrt{a + x}} \right) dx + C \), then \( f'(a) = \)

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{4} \)

Hint:

Differentiating Integrals with Parameters}
Differentiating an integral removes the integral
Simplify inside inverse trig using algebra
Remember \( \cos^{-1}\left( \frac{1}{\sqrt{2}} \right) = \frac{\pi}{4} \)

Answer 1

The Correct Option is D

Given: \[ f(x) = \int \cos^{-1} \left( \frac{x}{\sqrt{a + x}} \right) dx \Rightarrow f'(x) = \cos^{-1} \left( \frac{x}{\sqrt{a + x}} \right) \] Then: \[ f'(a) = \cos^{-1} \left( \frac{a}{\sqrt{a + a}} \right) = \cos^{-1} \left( \frac{a}{\sqrt{2a}} \right) = \cos^{-1} \left( \frac{1}{\sqrt{2}} \right) = \frac{\pi}{4} \]