Applications of Derivatives
Logarithmic Differentiation
Grade 12
Question:
<p>If \(y = x^{x^2}\), then \(\frac{dy}{dx}\) equals</p>
<p>(a) \(2\log x \cdot x^{x^2}\)</p>
<p>(b) \((2\log x + 1) \cdot x^{x^2}\)</p>
<p>(c) \((\log x + 1) \cdot x^{\frac{x^2-1}{2}}\)</p>
<p>(d) \(x^{x^2} \cdot \log(ex^2)\)</p>
Step-by-Step Solution
Key Concept: Use logarithmic differentiation for exponential functions where the base contains the variable.
<p><strong>Step 1:</strong> Use logarithmic differentiation: $\log y = x^2 \log x$</p><p><strong>Step 2:</strong> Differentiate both sides: $\frac{1}{y}\frac{dy}{dx} = 2x\log x + x^2 \cdot \frac{1}{x} = 2x\log x + x$</p><p><strong>Step 3:</strong> Therefore $\frac{dy}{dx} = y(2x\log x + x) = x^{x^2}(2\log x + 1)$</p><p>∴ Answer is (b).</p>
Correct Answer: B