Applications of Derivatives
Successive Derivatives
Grade 12
Question:
<p>If <span class='math'>e^y(x+1) = 1</span>, then <span class='math'>\frac{d^2y}{dx^2}</span> is equal to</p>
<p>(a) <span class='math'>y</span></p>
<p>(b) <span class='math'>-y</span></p>
<p>(c) <span class='math'>-y</span></p>
<p>(d) <span class='math'>\left(\frac{dy}{dx}\right)^2</span></p>
Step-by-Step Solution
Key Concept: Use implicit differentiation carefully and recognize relationships between variables.
<p><strong>Step 1:</strong> From <span class='math'>e^y(x+1) = 1</span>, we get <span class='math'>e^y = \frac{1}{x+1}</span></p><p><strong>Step 2:</strong> Differentiate: <span class='math'>e^y\frac{dy}{dx} = -\frac{1}{(x+1)^2}</span></p><p><strong>Step 3:</strong> Since <span class='math'>e^y = \frac{1}{x+1}</span>: <span class='math'>\frac{dy}{dx} = -\frac{1}{(x+1)^2} \cdot (x+1) = -\frac{1}{x+1}</span></p><p><strong>Step 4:</strong> Differentiate again: <span class='math'>\frac{d^2y}{dx^2} = \frac{1}{(x+1)^2}</span></p>
Correct Answer: C