Discussion
If \(e^{x}\left(y + 2\sqrt{1 + x}\right) = 5\), then \(-\frac{dy}{dx}\) at (0,3) is
Answer: D
\(-\frac{dy}{dx} =\) 6.
Differentiate \(e^{x}(y + 2\sqrt{1 + x}) = 5\) with the product rule:
\[ \begin{aligned} e^{x}(y + 2\sqrt{1 + x}) + e^{x}\left(y' + \frac{1}{\sqrt{1 + x}}\right) \\ &= 0 \end{aligned} \]
At \((0, 3)\): \(5 + y' + 1 = 0\), so \(y' = -6\) and \(-\frac{dy}{dx} = 6\). \(-6\) is \(\frac{dy}{dx}\) itself, the trap in the question's minus sign.
Implicit differentiation is the efficient route here, because solving for \(y\) first would introduce \(e^{-x}\) terms to differentiate.
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