Limits, Continuity & Differentiability
Differentiation
Grade 12
Question:
<p>If \(f(x) = \log_e x\) then the differential coefficient of \(f(\log_e x)\) with respect to \(x\) is</p>
<p>(a) \(\dfrac{\log_e x}{x}\)</p>
<p>(b) \(\dfrac{x}{\log_e x}\)</p>
<p>(c) \(\dfrac{1}{x \log_e x}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Use chain rule for composition: if f(x) = log_e(x), then f(log_e x) = log_e(log_e x). Differentiate by recognizing the outer function is logarithmic and inner function is also logarithmic.
<p><strong>Step 1:</strong> Identify the composition. Given f(x) = log_e(x), we need to find d/dx[f(log_e x)] = d/dx[log_e(log_e x)]</p><p><strong>Step 2:</strong> Apply chain rule. Let u = log_e x, then we need d/dx[log_e(u)] = (1/u) · du/dx</p><p><strong>Step 3:</strong> Calculate du/dx = d/dx(log_e x) = 1/x</p><p><strong>Step 4:</strong> Combine using chain rule: d/dx[log_e(log_e x)] = (1/log_e x) · (1/x) = 1/(x·log_e x)</p><p>∴ Answer: <strong>1/(x·log_e x)</strong> or <strong>1/(x·ln x)</strong></p>
Correct Answer: C