Limits, Continuity & Differentiability
Differentiation of infinite nested radicals
Grade 12
Question:
<p><strong>198.</strong> If \(y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}\), then the value of \(\dfrac{dy}{dx}\) at \(x = 8\) is:</p>
<p>(a) \(\dfrac{-4}{3}\)</p>
<p>(b) \(\dfrac{1}{12}\)</p>
<p>(c) \(\dfrac{4}{3}\)</p>
<p>(d) \(\dfrac{-1}{12}\)</p>
Step-by-Step Solution
Key Concept: Recognize the infinite nested radical as a self-similar function where y satisfies y = √(x/y), leading to y² = x/y, which gives y³ = x. Then differentiate using implicit differentiation or direct substitution.
<p><strong>Step 1:</strong> Let y = √(x/√(x/√(x/√(...)))) </p><p>Since the expression is infinite and self-similar, the denominator inside the first radical is also equal to y.</p><p><strong>Step 2:</strong> Set up the functional equation: y = √(x/y)</p><p><strong>Step 3:</strong> Square both sides: y² = x/y</p><p><strong>Step 4:</strong> Multiply by y: y³ = x</p><p><strong>Step 5:</strong> Differentiate implicitly with respect to x: 3y² · dy/dx = 1</p><p><strong>Step 6:</strong> Solve for dy/dx: dy/dx = 1/(3y²)</p><p><strong>Step 7:</strong> Find y when x = 8: y³ = 8 ⟹ y = 2</p><p><strong>Step 8:</strong> Substitute y = 2: dy/dx|ₓ₌₈ = 1/(3 · 2²) = 1/(3 · 4) = 1/12</p><p>∴ Answer: C</p>
Correct Answer: C