Yo dawg, I heard you like partial derivatives

Calculus Level 4

Let a function \(f\) in three variables \(x,y,z\) be defined as follows :


Also, we define the following :

\[\LARGE \mathcal{E}_n(x,y,z)=\sum_{i\in\{x,y,z\}}f_{\underbrace{iii\ldots ii}_{n\textrm{ times}}}~\forall~n\in\Bbb{Z^+}\]

If the value of \(\mathcal{E}_{101}(1,0,0)\) can be represented as \(A\cdot B!\) for integers \(A,B\) where \(A\) is a prime, then submit your answer as the value of \((A+B)\).

Bonus: Find the explicit formula(s) for \(\mathcal{E}_n(x,y,z)\) where \(n\in\Bbb{Z^+}\).

This problem is inspired by one of my calculus professors.

Notations used :

  • For a function \(f\) of \(n\) variables \(x_1,x_2,\ldots,x_n\), we denote \(\dfrac{\partial f}{\partial x_i}\) as the partial derivative of \(f\) w.r.t \(x_i\) where \(i\in\{1,2,\ldots,n\}\).

  • \(f_{a_1a_2\ldots a_n}=\dfrac{\partial}{\partial a_1}\left(\dfrac{\partial}{\partial a_2}\left(\cdots\left(\dfrac{\partial f}{\partial a_n}\right)\cdots\right)\right)\) where \(f\) is a function of \(n\) variables \(a_1,a_2,\ldots,a_n\).

  • \(\ln(\cdot)\) denotes the natural logarithm.


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