Limits, Continuity & Differentiability
Differentiation Rules
Grade 12
Question:
<p>Consider \(f(x) = x \ln x\) and \(g(x) = e^{2x}\). Let \(a\) and \(b\) be two values of \(x\) satisfying \(f(x) = g(x)\) with \(a < b\).</p><p>If \(h(x) = \frac{f(x)}{g(x)}\), then \(h'(a)\) equals:</p>
<p>(a) \(e\)</p>
<p>(b) \(-e\)</p>
<p>(c) \(3e\)</p>
<p>(d) \(-3e\)</p>
Step-by-Step Solution
Key Concept: Use the quotient rule to differentiate $h(x) = \frac{f(x)}{g(x)}$. Compute $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ and evaluate at $x = a$ using $f(a) = g(a)$.
<p>Answer: (d)</p>
Correct Answer: D