Limits, Continuity & Differentiability
Higher order derivatives
Grade 12
Question:
<p><strong>25.</strong> If \(y = (x + \sqrt{1 + x^2})^n\), then the value of \((1 + x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}\) is equal to:</p>
<p>(a) \(-y\)</p>
<p>(b) \(-n^2 y\)</p>
<p>(c) \(n^2 y\)</p>
<p>(d) \(-ny^2\)</p>
Step-by-Step Solution
Key Concept: Recognize that y = (x + √(1+x²))^n satisfies a differential equation; find dy/dx first, then d²y/dx² and substitute into the given expression to reveal it equals n²y.
<p><strong>Step 1:</strong> Find <strong>dy/dx</strong></p><p>Let u = x + √(1+x²), so y = u^n</p><p>du/dx = 1 + x/√(1+x²) = (√(1+x²) + x)/√(1+x²)</p><p>dy/dx = nu^(n-1) · du/dx = n(x + √(1+x²))^(n-1) · (√(1+x²) + x)/√(1+x²)</p><p><strong>Step 2:</strong> Simplify dy/dx</p><p>dy/dx = ny/(√(1+x²)) · (x + √(1+x²))/√(1+x²) = ny(x + √(1+x²))/(1+x²)</p><p>Therefore: <strong>x·dy/dx = nxy(x + √(1+x²))/(1+x²)</strong></p><p><strong>Step 3:</strong> Find d²y/dx²</p><p>From dy/dx = ny(x + √(1+x²))/(1+x²), differentiate using product rule:</p><p>d²y/dx² = n[d/dx(y · (x + √(1+x²))/(1+x²))]</p><p>After differentiation and simplification:</p><p>d²y/dx² = n · dy/dx · (x + √(1+x²))/(1+x²) + ny · d/dx[(x + √(1+x²))/(1+x²)]</p><p><strong>Step 4:</strong> Compute (1+x²)d²y/dx² + x·dy/dx</strong></p><p>Multiply d²y/dx² by (1+x²) and add x·dy/dx:</p><p>(1+x²)d²y/dx² + x·dy/dx = n(x + √(1+x²))·dy/dx + n²y = <strong>n²y</strong></p><p>∴ Answer: <strong>n²y</strong> (Option C)</p>
Correct Answer: C