Limits, Continuity & Differentiability
Higher order derivatives
Grade 12
Question:
<p>If \(y^2 = P(x)\) which is a polynomial of degree 3, then \(2\dfrac{d}{dx}\left(y^3 \dfrac{d^2y}{dx^2}\right)\) equals</p>
<p>(a) \(P''(x) + P'(x)\)</p>
<p>(b) \(P''(x) \cdot P'''(x)\)</p>
<p>(c) \(P(x) \cdot P''(x)\)</p>
<p>(d) a constant</p>
Step-by-Step Solution
Key Concept: Differentiate y² = P(x) twice to establish relationships between y and its derivatives, then use these to simplify the given expression. The key is recognizing that y³(d²y/dx²) can be expressed in terms of P(x) and its derivatives through systematic differentiation.
<p><strong>Step 1:</strong> Given y² = P(x) where P(x) is degree 3. Differentiate once:</p><p>2y(dy/dx) = P'(x), so dy/dx = P'(x)/(2y)</p><p><strong>Step 2:</strong> Differentiate again to find d²y/dx²:</p><p>2(dy/dx)² + 2y(d²y/dx²) = P''(x)</p><p>d²y/dx² = [P''(x) - 2(dy/dx)²]/(2y) = [P''(x) - P'(x)²/(2y²)]/(2y)</p><p><strong>Step 3:</strong> Multiply by y³:</p><p>y³(d²y/dx²) = y²[P''(x) - P'(x)²/(2y²)]/2 = [P(x)P''(x) - P'(x)²/2]/2</p><p><strong>Step 4:</strong> Differentiate with respect to x:</p><p>d/dx[y³(d²y/dx²)] = d/dx[(1/2)P(x)P''(x) - (1/4)P'(x)²]</p><p>= (1/2)[P'(x)P''(x) + P(x)P'''(x)] - (1/2)P'(x)P''(x)</p><p>= (1/2)P(x)P'''(x)</p><p><strong>Step 5:</strong> Multiply by 2:</p><p>2 · d/dx[y³(d²y/dx²)] = P(x)P'''(x)</p><p>∴ Answer: C</p>
Correct Answer: C