Question

If \[ y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots\infty}}}, \] then \[ \frac{dy}{dx}= \]

• A. \(\frac{1}{2y}\)
• B. \(\frac{1}{1-2y}\)
• C. \(\frac{1}{2(1-2y)}\)
• D. \(\frac{-1}{1-2y}\)

Hint:

For infinite radicals, replace the repeated radical part by the original variable expression.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
This problem deals with an infinite nested square root function. Because the expression is recursive and extends to infinity, the part inside the first square root (starting from the second \( x \)) is identical to the original function \( y \).
This allows us to convert the infinite expression into a finite algebraic equation.
Step 2: Key Formula or Approach:
1. Replace the infinite recursive part with \( y \).
2. Square both sides to eliminate the outer radical.
3. Use implicit differentiation to find \( dy/dx \).
Step 3: Detailed Explanation:

Given: \( y = \sqrt{x + \sqrt{x + \sqrt{x + \dots}}} \)

Since the pattern continues infinitely, the term under the first radical is \( x + y \).
\[ y = \sqrt{x + y} \]

Square both sides to remove the root:
\[ y^2 = x + y \]

Differentiate both sides with respect to \( x \) implicitly:
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(x + y) \]
\[ 2y \frac{dy}{dx} = 1 + \frac{dy}{dx} \]

Collect the terms containing \( dy/dx \) on one side:
\[ 2y \frac{dy}{dx} - \frac{dy}{dx} = 1 \]
\[ \frac{dy}{dx} (2y - 1) = 1 \]

Solve for \( dy/dx \):
\[ \frac{dy}{dx} = \frac{1}{2y - 1} \]

To match the given options, we can multiply the numerator and denominator by -1:
\[ \frac{dy}{dx} = \frac{-1}{1 - 2y} \]

This corresponds to Option (D).

Step 4: Final Answer:
The derivative \( dy/dx \) is \( \frac{-1}{1-2y} \).