Limits, Continuity & Differentiability
Higher Order Derivatives
Grade 12
Question:
<p>Given \(e^y + xy = e\). If \(y = f(x)\), then find \(\left(\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\right)\) at \(x = 0\).</p>
<p>\(\left(-\dfrac{1}{e}, \dfrac{1}{e^2}\right)\)</p>
<p>\(\left(\dfrac{1}{e}, \dfrac{1}{e^2}\right)\)</p>
<p>\(\left(-\dfrac{1}{e}, -\dfrac{1}{e^2}\right)\)</p>
<p>\(\left(\dfrac{1}{e}, -\dfrac{1}{e^2}\right)\)</p>
Step-by-Step Solution
Key Concept: Use implicit differentiation on e^y + xy = e, then find y(0) from the equation, dy/dx at x=0 using the first derivative, and d²y/dx² by differentiating the first derivative equation again.
<p><strong>Step 1: Find y at x = 0</strong></p><p>From e^y + xy = e, substitute x = 0:</p><p>e^y + 0 = e ⟹ y(0) = 1</p><p><strong>Step 2: Find dy/dx using implicit differentiation</strong></p><p>Differentiate e^y + xy = e with respect to x:</p><p>e^y · (dy/dx) + y + x(dy/dx) = 0</p><p>(e^y + x)(dy/dx) = -y</p><p>dy/dx = -y/(e^y + x)</p><p>At x = 0, y = 1: dy/dx = -1/(e + 0) = <strong>-1/e</strong></p><p><strong>Step 3: Find d²y/dx² by differentiating dy/dx equation</strong></p><p>Differentiate (e^y + x)(dy/dx) = -y with respect to x:</p><p>(e^y · dy/dx + 1)(dy/dx) + (e^y + x)(d²y/dx²) = -(dy/dx)</p><p>At x = 0, y = 1, dy/dx = -1/e:</p><p>(e · (-1/e) + 1)(-1/e) + (e + 0)(d²y/dx²) = -(-1/e)</p><p>(0)(-1/e) + e(d²y/dx²) = 1/e</p><p>d²y/dx² = <strong>1/e²</strong></p><p><strong>∴ Answer: (dy/dx, d²y/dx²) = (-1/e, 1/e²)</strong></p>
Correct Answer: A