Method of Differentiation
Infinite Series
Premium Question
Grade 12
Question:
If $y = x^{x^{x^{\dots}}}$ (an infinite tower, convergent), then $\frac{dy}{dx}$ equals:
(1) $\frac{y^2}{x(1 - y \ln x)}$
(2) $\frac{y}{x(1 - y \ln x)}$
(3) $\frac{y^2}{x(1 + y \ln x)}$
(4) $\frac{y}{1 - y \ln x}$
Step-by-Step Solution
Key Concept: Use the self-repeating structure to write the relation compactly as $y = x^y$, then take logarithms to get $\ln y = y \ln x$ before differentiating implicitly.
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Correct Answer: (1)