Limits, Continuity & Differentiability
Implicit Differentiation
Grade 12
Question:
<p>If <span>\(y^x = e^{y-x}\)</span>, then <span>\(\frac{dy}{dx}\)</span> is equal to</p>
<p>(a) <span>\(\frac{1+\log y}{y\log y}\)</span></p>
<p>(b) <span>\(\frac{(1+\log y)^2}{y\log y}\)</span></p>
<p>(c) Option not fully visible in source</p>
<p>(d) Option not fully visible in source</p>
Step-by-Step Solution
Key Concept: Implicit differentiation combined with logarithmic form allows separation of variables.
<p><strong>Solution:</strong> Take logarithm of both sides: <span>$x \log y = y - x$</span>. Differentiate implicitly with respect to <span>$x$</span>: <span>$\log y + x \cdot \frac{1}{y} \cdot \frac{dy}{dx} = \frac{dy}{dx} - 1$</span>. Solve for <span>$\frac{dy}{dx}$</span>.</p>
Correct Answer: a