Method of Differentiation
Infinite Series
Premium Question
Grade 12
Question:
If $y = \sqrt{x + \sqrt{x + \sqrt{x + \dots \infty}}}$, then $\frac{dy}{dx}$ equals:
(1) $\frac{1}{2y - 1}$
(2) $\frac{1}{2y + 1}$
(3) $\frac{1}{2y}$
(4) $\frac{2y}{2y - 1}$
Step-by-Step Solution
Key Concept: Use the self-similar structure to write $y^2 = x + y$, which converts the nested radical into a simple algebraic relation that can be differentiated implicitly.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)