Applications of Derivatives
Logarithmic Differentiation
Grade 12
Question:
<p>If <span>\(y = x(\ln x)^{\ln(\ln x)}\)</span>, then <span>\(\frac{dy}{dx}\)</span> is equal to</p>
<p>(a) <span>\(\frac{y}{x}\left(\ln x + 2\ln x \ln(\ln x)\right)\)</span></p>
<p>(b) <span>\(\frac{y}{x}(\ln x)^{\ln(\ln x)}(2\ln(\ln x) + 1)\)</span></p>
<p>(c) <span>\(\frac{y}{x}\left((\ln x)^2 + 2\ln x \ln(\ln x)\right)\)</span></p>
<p>(d) <span>\(\frac{y}{x\ln x}(2\ln(\ln x) + 1)\)</span></p>
Step-by-Step Solution
Key Concept: Use logarithmic differentiation to handle the complex composite function.
<p>Let <span>$y = x(\ln x)^{\ln(\ln x)}$</span></p><p>Taking logarithm: <span>$\ln y = \ln x + \ln(\ln x) \cdot \ln(\ln x)$</span></p><p>Differentiating: <span>$\frac{1}{y}\frac{dy}{dx} = \frac{1}{x} + \frac{d}{dx}[\ln(\ln x) \cdot \ln(\ln x)]$</span></p><p><span>$\frac{1}{y}\frac{dy}{dx} = \frac{1}{x} + 2\ln(\ln x) \cdot \frac{1}{x\ln x}$</span></p><p><span>$\frac{dy}{dx} = \frac{y}{x}\left(\ln x + 2\ln x \ln(\ln x)\right)$</span></p>
Correct Answer: A