# Calculation of the derivative as a limit at a point

This online calculator performs а numerical differentiation - the approximate calculation of the derivative of a function at a given point. It calculates the derivative as a value of the limit at a given point by successive approximations until the specified accuracy is reached.

A description of the method for calculating the value of the derivative can be found below the calculator.

#### Calculation of the derivative as a limit at a point

Derivative of the function

Digits after the decimal point: 4
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### Calculating the derivative by its definition as a limit

The problem of numerical differentiation arises when a function is given tabularly, or when direct differentiation is difficult (for example, when the analytic form of the function is complex). Yet, if the function is given analytically, we can apply the calculation of the value of the derivative by its definition as a limit.

In this case, we have some function $y=f(x)$ for which we need to calculate the value of the derivative at the point x₀. We assume that this function is defined in the neighborhood of the point x₀ and has a derivative at this point. Based on the definition of the derivative
$y'(x_0) = f'(x_0)=\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$
there exists a limit of the ratio of the increment of the function Δy to the increment of the argument Δx at Δx→0, where
$\Delta x = x - x_0 \ \ \Delta y = f(x_0 + \Delta x) - f(x_0)$

The value of the derivative can be obtained by going to the limit in progressively smaller steps until the required accuracy is reached. For this purpose, at each step of the n sequence, the increment of the argument is calculated using the following formula
$\Delta x = \Delta x_n = \frac {\Delta x_0}{a^n}$,
where
Δx₀ is the initial increment of the argument, e.g., 0.1
a is some number greater than 1, e.g., 10
n = 0, 1, ...

Then
$y'(x_0) \approx \frac{\Delta y_n}{\Delta x_n}$

The sequence stops when the following condition is met
$\frac{\Delta y_n}{\Delta x_n} - \frac{\Delta y_{n-1}}{\Delta x_{n-1}}} \le \epsilon$

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