Step 1: Understanding the Concept:
This is an infinite series function. Since the pattern repeats infinitely, we can replace the inner part of the square root with \(y\) itself to create a finite algebraic equation. Step 2: Key Formula or Approach:
1. Recursive substitution: \(y = \sqrt{x + y}\).
2. Implicit differentiation: Differentiate both sides with respect to \(x\). Step 3: Detailed Explanation:
Given \( y = \sqrt{x + y} \).
Square both sides:
\[ y^2 = x + y \]
Differentiate with respect to \(x\) using the chain rule:
\[ 2y \frac{dy}{dx} = 1 + \frac{dy}{dx} \]
Rearrange to group \(\frac{dy}{dx}\) terms:
\[ 2y \frac{dy}{dx} - \frac{dy}{dx} = 1 \]
\[ \frac{dy}{dx} (2y - 1) = 1 \]
\[ \frac{dy}{dx} = \frac{1}{2y - 1} \]
Note: \(\frac{1}{2y-1}\) is mathematically identical to \(\frac{-1}{1-2y}\). Step 4: Final Answer:
The derivative is \( \frac{1}{2y - 1} \).