Exponentiated Geometric Sum

Calculus Level 2

\[ \large y_n (x) = e^x \times e^{x^2} \times e^{x^3} \times \ldots \times e^{x^n} \]

For some positive integer \(n\), let \(y_n(x) \) denote function of \(x\) as stated above.

What is the value of the limit below?

\[ \large \lim_{n\to\infty} \left . \frac d{dx} \left ( y_n (x) \right ) \right |_{x=\frac12} \]

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