<p>If \(y = e^{nx}\), then find the value of \(\left(\frac{d^2 y}{dx^2}\right)\left(\frac{d^2 x}{dy^2}\right)\).</p>
Step-by-Step Solution
Key Concept: You must find both second derivatives separately using the chain rule and inverse function differentiation, then recognize that d²x/dy² requires treating x as a function of y, not simply taking reciprocals of derivatives.
<p><strong>Step 1: Find d²y/dx²</strong></p><p>Given: y = e^(nx)</p><p>dy/dx = ne^(nx)</p><p>d²y/dx² = n²e^(nx)</p><p><strong>Step 2: Find d²x/dy² (treating x as function of y)</strong></p><p>From y = e^(nx), we have: x = (ln y)/n</p><p>dx/dy = 1/(ny)</p><p>d²x/dy² = d/dy[1/(ny)] = -1/(ny²)</p><p><strong>Step 3: Find the product</strong></p><p>(d²y/dx²)(d²x/dy²) = n²e^(nx) × [-1/(ne^(nx))²]</p><p>= n²e^(nx) × [-1/(n²e^(2nx))]</p><p>= -1/e^(nx)</p><p><strong>Step 4: Simplify using y = e^(nx)</strong></p><p>(d²y/dx²)(d²x/dy²) = -1/y</p><p>∴ Answer: <strong>-1/y</strong> or equivalently <strong>-e^(-nx)</strong></p>
Correct Answer: A