<p>If \(x = e^t \cos t\), \(y = e^t \sin t\) then \(\dfrac{d^2 y}{dx^2}\) = ______</p>
Step-by-Step Solution
Key Concept: Use parametric differentiation: find dy/dt and dx/dt separately, then apply the chain rule twice. For the second derivative, use d²y/dx² = (d/dt(dy/dx))/(dx/dt).
<p><strong>Step 1:</strong> Find dy/dt and dx/dt</p><p>x = e^t cos t → dx/dt = e^t cos t - e^t sin t = e^t(cos t - sin t)</p><p>y = e^t sin t → dy/dt = e^t sin t + e^t cos t = e^t(sin t + cos t)</p><p><strong>Step 2:</strong> Find dy/dx</p><p>dy/dx = (dy/dt)/(dx/dt) = e^t(sin t + cos t)/[e^t(cos t - sin t)] = (sin t + cos t)/(cos t - sin t)</p><p><strong>Step 3:</strong> Find d(dy/dx)/dt using quotient rule</p><p>Let u = sin t + cos t, v = cos t - sin t</p><p>du/dt = cos t - sin t, dv/dt = -sin t - cos t</p><p>d(dy/dx)/dt = [(cos t - sin t)(cos t - sin t) - (sin t + cos t)(-sin t - cos t)]/[(cos t - sin t)²]</p><p>= [(cos t - sin t)² + (sin t + cos t)²]/[(cos t - sin t)²]</p><p>= [cos²t - 2sin t cos t + sin²t + sin²t + 2sin t cos t + cos²t]/[(cos t - sin t)²]</p><p>= 2/[(cos t - sin t)²]</p><p><strong>Step 4:</strong> Find d²y/dx²</p><p>d²y/dx² = [d(dy/dx)/dt]/(dx/dt) = {2/[(cos t - sin t)²]}/{e^t(cos t - sin t)}</p><p>= 2/[e^t(cos t - sin t)³]</p><p>∴ Answer: <strong>2e⁻ᵗ/(cos t - sin t)³</strong> or equivalently <strong>2/[eᵗ(cos t - sin t)³]</strong></p>
Correct Answer: 2