# Calculating the partial derivative by its definition

This online calculator performs numerical differentiation of a function of several variables - the approximate calculation of all partial derivatives of a function at a given point - over all variables.

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Created: 2023-12-14 10:36:31, Last updated: 2023-12-17 21:49:57

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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).

#### Calculating the partial derivative by its definition

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