Limits, Continuity & Differentiability
Differentiation under integral sign
Grade 12
Question:
<p>Let \(f(x) = \displaystyle\int_0^x t\ln(1+t^2)\, dt\), 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 for differentiation under the integral sign: f'(x) = x·ln(1+x²), then differentiate again using the product rule to find f''(0).
<p><strong>Step 1:</strong> Apply Leibniz rule to find f'(x).</p><p>Since f(x) = ∫₀ˣ t·ln(1+t²) dt, by the Fundamental Theorem of Calculus:</p><p>f'(x) = x·ln(1+x²)</p><p><strong>Step 2:</strong> Differentiate f'(x) using the product rule to find f''(x).</p><p>f''(x) = d/dx[x·ln(1+x²)]</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 at x = 0.</p><p>f''(0) = ln(1+0) + 2(0)²/(1+0)</p><p>f''(0) = ln(1) + 0 = 0</p><p>∴ Answer: A (f''(0) = 0)</p>
Correct Answer: A