Exponentiated Geometric Sum

Calculus Level 2

yn(x)=ex×ex2×ex3××exn \large y_n (x) = e^x \times e^{x^2} \times e^{x^3} \times \ldots \times e^{x^n}

For some positive integer nn, let yn(x)y_n(x) denote function of xx as stated above.

What is the value of the limit below?

limnddx(yn(x))x=12 \large \lim_{n\to\infty} \left . \frac d{dx} \left ( y_n (x) \right ) \right |_{x=\frac12}

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