Definite Integration
Differentiation under integral sign / Leibniz rule
Grade 12
Question:
<p>If \(\displaystyle\int_0^x f(t)\, dt = x^2 + \int_x^1 t^2 f(t)\, dt\), then \(f'(1/2)\) is:</p>
<p>\(\dfrac{24}{25}\)</p>
<p>\(\dfrac{18}{25}\)</p>
<p>\(\dfrac{4}{5}\)</p>
<p>\(\dfrac{6}{25}\)</p>
Step-by-Step Solution
Key Concept: Differentiate both sides of the functional equation with respect to x using Leibniz rule, then substitute specific values to find f(x) and compute f'(1/2).
<p><strong>Step 1:</strong> Differentiate both sides with respect to x.</p><p>Given: ∫₀ˣ f(t)dt = x² + ∫ₓ¹ t²f(t)dt</p><p>Differentiating: f(x) = 2x - x²f(x)</p><p><strong>Step 2:</strong> Solve for f(x).</p><p>f(x) + x²f(x) = 2x</p><p>f(x)(1 + x²) = 2x</p><p>f(x) = 2x/(1 + x²)</p><p><strong>Step 3:</strong> Find f'(x) using quotient rule.</p><p>f'(x) = [(1 + x²)·2 - 2x·2x]/(1 + x²)²</p><p>f'(x) = [2 + 2x² - 4x²]/(1 + x²)²</p><p>f'(x) = [2 - 2x²]/(1 + x²)² = 2(1 - x²)/(1 + x²)²</p><p><strong>Step 4:</strong> Evaluate at x = 1/2.</p><p>f'(1/2) = 2(1 - 1/4)/(1 + 1/4)²</p><p>f'(1/2) = 2(3/4)/(5/4)²</p><p>f'(1/2) = (3/2)/(25/16) = (3/2)·(16/25) = 24/25</p><p>∴ Answer: A</p>
Correct Answer: A