Limits, Continuity & Differentiability
Differentiation using Leibniz rule
Grade 12
Question:
<p>Let \(f(x) = \int_0^x t\ln(1+t^2)\, dt\), then \(f''(0)\) is:</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Use Leibniz rule (differentiation under integral sign) to find f'(x), then differentiate again to find f''(x). The second derivative at x=0 requires evaluating the derivative of the integrand at t=0.
<p><strong>Step 1:</strong> Find f'(x) using Leibniz rule (fundamental theorem of calculus).</p><p>f(x) = ∫₀ˣ t·ln(1+t²) dt</p><p>By Leibniz rule: f'(x) = x·ln(1+x²)</p><p><strong>Step 2:</strong> Find f''(x) by differentiating f'(x).</p><p>f'(x) = x·ln(1+x²)</p><p>Using product rule:</p><p>f''(x) = 1·ln(1+x²) + x·(2x)/(1+x²)</p><p>f''(x) = ln(1+x²) + 2x²/(1+x²)</p><p><strong>Step 3:</strong> Evaluate f''(0).</p><p>f''(0) = ln(1+0) + 2(0)²/(1+0)</p><p>f''(0) = ln(1) + 0</p><p>f''(0) = 0</p><p>∴ Answer: A (which is 0)</p>
Correct Answer: A