Limits, Continuity & Differentiability
Implicit Differentiation
Grade 12
Question:
<p>If <span class='math'>x\log_e(\log_e x) - x^2 + y^2 = 4</span> <span class='math'>(y > 0)</span>, then <span class='math'>\frac{dy}{dx}</span> at <span class='math'>x = e</span> is equal to</p>
<p>(a) <span class='math'>\frac{e}{2}</span></p>
<p>(b) <span class='math'>\frac{1+2e}{2\sqrt{4+e}}</span></p>
<p>(c) <span class='math'>\frac{1+2e}{2\sqrt{4+e}}</span></p>
<p>(d) <span class='math'>\frac{2e-1}{2\sqrt{4+e}}</span></p>
Step-by-Step Solution
Key Concept: Use implicit differentiation and logarithmic function properties at a specific point.
<p><strong>Step 1:</strong> Differentiate implicitly with respect to <span class='math'>x</span>: <span class='math'>\log_e(\log_e x) + 1 - 2x + 2y\frac{dy}{dx} = 0</span></p><p><strong>Step 2:</strong> At <span class='math'>x = e</span>: <span class='math'>\log_e(1) + 1 - 2e + 2y\frac{dy}{dx} = 0</span></p><p><strong>Step 3:</strong> Find <span class='math'>y</span> at <span class='math'>x = e</span> from the original equation: <span class='math'>0 - e^2 + y^2 = 4 \Rightarrow y = \sqrt{4 + e^2}</span></p><p><strong>Step 4:</strong> Solve for <span class='math'>\frac{dy}{dx} = \frac{2e-1}{2\sqrt{4+e^2}}</span></p>
Correct Answer: B