Limits, Continuity & Differentiability
Logarithmic Differentiation
Grade 12
Question:
<p>Derivative of <span>\((\log x)^{\log x}\)</span> with respect to <span>\(x\)</span> is</p>
<p>(a) <span>\((\log x)^{\log x}\left\{\frac{1}{x} + \frac{\log(\log x)}{x}\right\}\)</span></p>
<p>(b) <span>\((\log x)^{\log x}\left\{\frac{\log(\log x)}{x} + \frac{1}{x}\right\}\)</span></p>
<p>(c) <span>\((\log x)^{\log x}\left\{\frac{\log x}{x} + \frac{\log(\log x)}{x}\right\}\)</span></p>
<p>(d) <span>\((\log x)^{\log x}\left\{\frac{1}{x} + \frac{\log(\log x)}{x}\right\}\)</span></p>
Step-by-Step Solution
Key Concept: Logarithmic differentiation is essential for functions of the form $f(x)^{g(x)}$.
<p><strong>Solution:</strong> Use logarithmic differentiation. Let <span>$y = (\log x)^{\log x}$</span>. Taking logarithm: <span>$\ln y = \log x \cdot \ln(\log x)$</span>. Differentiate both sides using product rule.</p>
Correct Answer: a