Question

If \( g(x) = -\sqrt{25 - x^2} \), then \( g'(1) \) is:

• A. \( -\sqrt{24} \)
• B. \( \sqrt{24} \)
• C. \( \frac{1}{24} \)
• D. \( \frac{1}{\sqrt{24}} \)
• E. \( \frac{-1}{\sqrt{24}} \)

Hint:

To differentiate square roots, use the chain rule: \( \frac{d}{dx} \left( \sqrt{f(x)} \right) = \frac{f'(x)}{2\sqrt{f(x)}} \).

Answer 1

The Correct Option is D

First, find the derivative of the function \( g(x) = -\sqrt{25 - x^2} \) using the chain rule. 
We have: \[ g'(x) = - \frac{d}{dx} \left( \sqrt{25 - x^2} \right) \] Using the chain rule: \[ g'(x) = - \frac{1}{2\sqrt{25 - x^2}} \cdot (-2x) = \frac{x}{\sqrt{25 - x^2}} \] Now, substitute \( x = 1 \): \[ g'(1) = \frac{1}{\sqrt{25 - 1^2}} = \frac{1}{\sqrt{24}} = \frac{1}{\sqrt{24}} \]