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 nested radical as a self-similar infinite expression, then convert it to a functional equation y = √(x/y) by assuming convergence. Solve this equation to find y explicitly, then differentiate.
<p><strong>Step 1:</strong> Let y = √(x/√(x/√(x/√(···)))). Since the pattern repeats infinitely, the denominator inside the first square root is also y.</p><p>Therefore: y = √(x/y)</p><p><strong>Step 2:</strong> Square both sides: y² = x/y</p><p>Multiply by y: y³ = x</p><p>Thus: y = x^(1/3)</p><p><strong>Step 3:</strong> Differentiate y = x^(1/3) with respect to x:</p><p>dy/dx = (1/3)x^(-2/3) = 1/(3x^(2/3))</p><p><strong>Step 4:</strong> Evaluate at x = 8:</p><p>dy/dx|_(x=8) = 1/(3·8^(2/3)) = 1/(3·(2³)^(2/3)) = 1/(3·2²) = 1/(3·4) = 1/12</p><p>∴ Answer: C</p>
Correct Answer: C