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Solve the following.
If x = e^(y + ey + ey + … ∞), x > 0 then find (dy)/(dx).
If $x = e^{y + e^{y + e^{y + \dots \infty}}}$, $x > 0$ then find $\frac{dy}{dx}$.
Answer
Answer
Since the exponent repeats, $x = e^{y + x}$.
Taking logarithms: $\ln x = y + x$, so $y = \ln x - x$.
Differentiating: $\frac{dy}{dx} = \frac{1}{x} - 1$.
Final answer: $\frac{1-x}{x}$