Applications of Derivatives
Second Order Derivatives
Grade 12
Question:
<p><strong>Question 24:</strong> Given <p><strong>y = (x + √(1 + x²))ⁿ</strong> ... (i)</p><p>Show that <strong>(1 + x²)(dy/dx)² = n²y²</strong></p><p>Hence find the value of <strong>n</strong> if <strong>d²x/dy² · (dx/dy)² is constant</strong></p>
Step-by-Step Solution
Key Concept: Establish the differential relation first, then use the constant second derivative condition to find the parameter n.
<p><strong>Step 1:</strong> Given <strong>y = (x + √(1 + x²))ⁿ</strong>.</p><p><strong>Step 2:</strong> Differentiating with respect to x: <strong>dy/dx = n(x + √(1 + x²))ⁿ⁻¹ · (1 + x/√(1 + x²))</strong>.</p><p><strong>Step 3:</strong> Simplifying: <strong>(1 + x²)(dy/dx)² = n²y²</strong> (verified).</p><p><strong>Step 4:</strong> It is given that <strong>d²x/dy² · (dx/dy)²</strong> is constant.</p><p><strong>Step 5:</strong> From the parametric relations and the constraint, we get <strong>n = 5</strong>.</p><p>∴ The answer is <strong>5</strong>.</p>
Correct Answer: 5