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For a function of a single variable f(x), the derivative at x0 is the slope or the tangent at this point.

For the real function f, if (f(x+x0) - f(x0)) / (x-x0) has a limit as x approaches the original point x0, this limit is the derivative of f at x0 and is denoted by f '(x0).

this value is useful for scientists in order to obtain the rate of change of some variable of interest in dynamical systems.

- To illustrate graphically the derivative of a function.
- To define the differential of a function.