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.

A graphic illustration of the integral of a function.  You can change the limits of integration and the number of rectangles employed for the approximation. 

The derivative of a function at a point gives the slope at that point.