If y = e^{x + e^{x + e^{x + dots infty}}},the | vedclass

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Question

(A) Given the infinite series $y = e^{x + e^{x + e^{x + \dots \infty}}}$.Since the exponent repeats,we can write the equation as $y = e^{x + y}$.Takin

Vedclass · Accepted Answer

$\frac{y}{1 - y}$

Vedclass · Answer

(A) Given the infinite series $y = e^{x + e^{x + e^{x + \dots \infty}}}$.Since the exponent repeats,we can write the equation as $y = e^{x + y}$.Taking the natural logarithm on both sides,we get $\ln(y) = x + y$.Differentiating both sides with respect to $x$,we get $\frac{1}{y} \frac{dy}{dx} = 1 + \frac{dy}{dx}$.Rearranging the terms to solve for $\frac{dy}{dx}$,we have $\frac{1}{y} \frac{dy}{dx} - \frac{dy}{dx} = 1$.$\frac{dy}{dx} (\frac{1}{y} - 1) = 1$.$\frac{dy}{dx} (\frac{1 - y}{y}) = 1$.Therefore,$\frac{dy}{dx} = \frac{y}{1 - y}$.

If $y = e^{x + e^{x + e^{x + \dots \infty}}}$,then $\frac{dy}{dx} = $

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