Question

If \(e^{x}(y + 2\sqrt{1 + x}) = 5\), then \(-\frac{dy}{dx}\) at \((0,3)\) is

• A. 2
• B. -2
• C. -3
• D. 6
• E. -6

Hint:

When differentiating \(\sqrt{1+x}\), remember \(\frac{d}{dx}(1+x)^{1/2} = \frac{1}{2\sqrt{1+x}}\).

Answer 1

The Correct Option is D

Step 1: Concept:
• Use implicit differentiation.
• Differentiate both sides with respect to \(x\).

Step 2: Detailed Explanation:
• Given: \[ e^{x}(y + 2\sqrt{1+x}) = 5 \]
• Differentiate both sides: \[ e^{x}(y + 2\sqrt{1+x}) + e^{x}\left(\frac{dy}{dx} + \frac{1}{\sqrt{1+x}}\right) = 0 \]
• Substitute point \((0,3)\): \[ e^{0}(3 + 2\sqrt{1}) + e^{0}\left(\frac{dy}{dx} + \frac{1}{\sqrt{1}}\right) = 0 \]
• Simplify: \[ (3 + 2) + \left(\frac{dy}{dx} + 1\right) = 0 \]
• Solve: \[ 5 + \frac{dy}{dx} + 1 = 0 \Rightarrow \frac{dy}{dx} + 6 = 0 \Rightarrow \frac{dy}{dx} = -6 \]
• Hence: \[ -\frac{dy}{dx} = 6 \]

Step 3: Final Answer:
• \[ -\frac{dy}{dx} = 6 \]