What is a derivative?

A derivative is a mathematical description of how one quantity will change with respect to another. For example the 1st derivative of some function will give you the gradient of the tangent at every point on that function. In the context of mechancis say you had postion as a function of time the 1st derivative would give you the speed or velocity depending on how the function is specified. i.e velocity/speed is the rate of change of postion with respect to time.