Limits, Continuity & Differentiability
Differentiation of implicit/infinite expressions
Grade 12
Question:
<p><strong>320.</strong> If \(y = \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}}\), where \(x > 0\), then \(\dfrac{dy}{dx}\) can be:</p>
<p>(a) \(\dfrac{1}{2y-1}\)</p>
<p>(b) \(\dfrac{x}{x+2y}\)</p>
<p>(c) \(\dfrac{1}{\sqrt{1+4x}}\)</p>
<p>(d) \(\dfrac{y}{2x+y}\)</p>
Step-by-Step Solution
Key Concept: Recognize that the nested radical converges to a limit y, then use the self-similar property y = √(x + y) to establish a functional equation and differentiate implicitly.
<p><strong>Step 1:</strong> Since y = √(x + √(x + √(x + ...))), the expression under the first radical is also y itself.</p><p>Therefore: <strong>y = √(x + y)</strong></p><p><strong>Step 2:</strong> Square both sides: <strong>y² = x + y</strong></p><p>Rearrange: <strong>y² - y - x = 0</strong></p><p><strong>Step 3:</strong> Solve for y using the quadratic formula (taking the positive root since y > 0):</p><p><strong>y = (1 + √(1 + 4x))/2</strong></p><p><strong>Step 4:</strong> Differentiate y with respect to x:</p><p>dy/dx = d/dx[(1 + √(1 + 4x))/2]</p><p>dy/dx = (1/2) · (1/2√(1 + 4x)) · 4</p><p><strong>dy/dx = 1/√(1 + 4x)</strong></p><p><strong>Step 5:</strong> Express in terms of y. From y² - y - x = 0: x = y² - y</p><p>So: 1 + 4x = 1 + 4(y² - y) = 4y² - 4y + 1 = (2y - 1)²</p><p>Therefore: <strong>dy/dx = 1/(2y - 1)</strong></p><p><strong>Step 6:</strong> Alternative form: Since 2y = 1 + √(1 + 4x), we have:</p><p><strong>dy/dx = √(1 + 4x)/(1 + √(1 + 4x))</strong> or equivalently <strong>1/(2y - 1)</strong></p><p>∴ Answer: A, C</p>
Correct Answer: A,C