Limits, Continuity & Differentiability
Differentiation
Grade 12
Question:
<p>If \(x^y = e^{x-y}\), then \(\dfrac{dy}{dx}\) is</p>
<p>\(\dfrac{1+x}{1+\log x}\)</p>
<p>\(\dfrac{1 - \log x}{1 + \log x}\)</p>
<p>not defined</p>
<p>\(\dfrac{\log x}{(1 + \log x)^2}\)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation on the equation x^y = e^(x-y) by taking natural logarithm of both sides, converting the transcendental equation into a form where logarithmic differentiation is straightforward.
<p><strong>Step 1:</strong> Take natural logarithm of both sides of x^y = e^(x-y)</p><p>ln(x^y) = ln(e^(x-y))</p><p>y·ln(x) = x - y</p><p><strong>Step 2:</strong> Differentiate both sides with respect to x using implicit differentiation</p><p>d/dx[y·ln(x)] = d/dx[x - y]</p><p>dy/dx·ln(x) + y·(1/x) = 1 - dy/dx</p><p><strong>Step 3:</strong> Collect dy/dx terms on one side</p><p>dy/dx·ln(x) + dy/dx = 1 - y/x</p><p>dy/dx[ln(x) + 1] = 1 - y/x</p><p><strong>Step 4:</strong> Solve for dy/dx</p><p>dy/dx = (1 - y/x)/(ln(x) + 1) = (x - y)/(x(ln(x) + 1))</p><p>∴ Answer: <strong>dy/dx = (x - y)/(x(1 + ln(x)))</strong> or equivalent form</p>
Correct Answer: D