Method of Differentiation
Infinite Series
Premium Question
Grade 12
Question:
If $y = e^{x+e^{x+e^{x+\dots \infty}}}$, then $\frac{dy}{dx}$ equals:
(1) $\frac{y}{1 - y}$
(2) $\frac{y}{1 + y}$
(3) $\frac{1 - y}{y}$
(4) $\frac{y^2}{1 - y}$
Step-by-Step Solution
Key Concept: Recognise that the infinite tower repeats itself, so the relation can be rewritten compactly as $y = e^{x+y}$, i.e. $\ln y = x + y$ — differentiate this implicitly.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)