<p>If \(y = e^{nx}\) then \(\left(\dfrac{d^2y}{dx^2}\right)\left(\dfrac{d^2x}{dy^2}\right)\) is equal to</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation to find d²x/dy² in terms of dy/dx, then multiply with d²y/dx² carefully accounting for the chain rule and reciprocal relationships.
<p><strong>Step 1:</strong> Find dy/dx and d²y/dx²</p><p>Given y = e^(nx), we have:</p><p>dy/dx = ne^(nx) = ny</p><p>d²y/dx² = n²e^(nx) = n²y</p><p><strong>Step 2:</strong> Find dx/dy and d²x/dy²</p><p>From dy/dx = ny, we get: dx/dy = 1/(ny)</p><p>Differentiating with respect to y:</p><p>d²x/dy² = d/dy[1/(ny)] = -1/(ny²) · n = -n/(n²y²) = -1/(ny²)</p><p><strong>Step 3:</strong> Multiply the second derivatives</p><p>(d²y/dx²)(d²x/dy²) = (n²y) · (-1/(ny²))</p><p>= -n²y/(ny²)</p><p>= -n/y</p><p>Since y = e^(nx):</p><p>(d²y/dx²)(d²x/dy²) = -n/e^(nx) = -ne^(-nx)</p><p>∴ Answer: D (which equals -ne^(-nx) or equivalent form)</p>
Correct Answer: D