Limits, Continuity & Differentiability
Differentiation of implicit functions
Grade 12
Question:
<p>Given that, \(x^y = e^{x-y}\). Find \(\dfrac{dy}{dx}\).</p>
<p>\(\dfrac{\ln x}{(1+\ln x)^2}\)</p>
<p>\(\dfrac{x - y}{x(1+\ln x)}\)</p>
<p>\(\dfrac{1}{1+\ln x}\)</p>
<p>\(\dfrac{y}{x(1+\ln x)}\)</p>
Step-by-Step Solution
Key Concept: Take natural logarithm of both sides to convert the exponential equation into a form suitable for implicit differentiation, then apply the chain rule carefully.
<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 the product rule and chain rule:</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 terms containing dy/dx 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)</p><p>dy/dx = (x - y)/(x(ln(x) + 1))</p><p>∴ Answer: <strong>dy/dx = (x - y)/(x(1 + ln(x)))</strong></p>
Correct Answer: A