# 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 :

$f(x,y,z)=\ln(x^3+y^3+z^3-3xyz)$

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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