Online calculator: Calculating the partial derivative by its definition

The function is given by an analytic expression, so to find the derivative we use the method of going to the limit by successive approximations until a given accuracy is reached, similar to the method used in Calculation of the derivative as a limit at a point to calculate the derivative of a function of one variable.
Defining the partial derivative of a function $f(x_1, x_2, ..., x_n)$ at a point $(a_1, a_2, ..., a_n)$ on a variable $x_k$ :

${\frac {\partial f}{\partial x_{k}}}(a_{1},\cdots ,a_{n})=\lim _{{\Delta x\to 0}}{\frac {f(a_{1},\ldots ,a_{k}+\Delta x_k,\ldots ,a_{n})-f(a_{1},\ldots ,a_{k},\ldots ,a_{n})}{\Delta x_k}}$

The calculator computes the value of the expression $\frac{\Delta y}{\Delta x_k}$ in constantly decreasing steps $\Delta x_k$ until the desired accuracy is reached. At each approximation $n (n = 0, 1, 2, ... )$ the incremental step of the variable $x_k$ decreases according to the rule $\Delta x_k = \Delta x_k_n = \frac {\Delta x_k_0}{a^n}$ , where the initial step $\Delta x_k_0$ and the parameter $a > 1$ can be set in the calculator (by default, the initial step is 0. 1 and the parameter a is 10).