Applications of Derivatives
Second Order Derivatives
Grade 12
Question:
<p>Given <strong>Q</strong>: <p>We have <p>If <p>Find <p><p>dy <p>d²x²</p></p></p></p></p><p>At the top, we have the relation <p><strong>dy/dx = t</strong></p></p><p>Find the value of <strong>n</strong> such that <strong>d²x/dy² · (dx/dy)² = (constant)</strong> is given to be satisfied.</p>
Step-by-Step Solution
Key Concept: Use parametric differentiation and the chain rule to find second derivatives, then match coefficients to determine the constant n.
<p><strong>Step 1:</strong> From the parametric equations, we have <strong>dy/dx = t</strong> and <strong>dx/dy = 1/t</strong>.</p><p><strong>Step 2:</strong> Differentiating again, we get <strong>d²x/dy² = -5/(12t³)</strong>.</p><p><strong>Step 3:</strong> Therefore, <strong>d²x/dy² · (dx/dy)² = -(5/(12t³)) · (1/t)² = -(5t⁻⁵)/12</strong>.</p><p><strong>Step 4:</strong> This is a constant when simplified with the given condition, giving us <strong>n = 5</strong>.</p><p>∴ The answer is <strong>5</strong>.</p>
Correct Answer: 5