Derivative (calculus)
In mathematics a derivative is the change in the function with respect to one of it vaiables. Essentially, the derivative is a means to calculate the gradient or rate of change at a particular value for a given function, f. Consequently, it can be used to calculate velocity from a displacement-time graph, or acceleration from a velocity-time graph. Furthermore, it can be used to calculate the rate of cooling from a temperature-time graph.
The process of finding a derivative is called differentiation.
Algebraic differentiation is an important part of calculus, an essential branch of mathematics.
For a single variable real function the derivative is the equation that given the slope of the line which is tangential at that point.
When defined from the first principals, the derivative of a function is the limit of the average rate of change of the function over 
as 
tends to zero. In other words, the derivative
provided the above limit exists.
Alternative notation also commonly found is 
.
For example, a polynomial can be differentiated by taking into account the linearity of the derivative, and by using the general formula:
(Proving this is a worth while exercise).
For example, if 
, the derivative with respect to 
is
Contents
Properties of the derivative
is a proper quotient and a result,
is a valid operation and is much used in solving differential equations.
The differential operator has an associated eigenfunction,
where e is the constant defined for this purpose.
Important differentiation rules
Higher order derivatives
A higher order derivative is obtained by repeating derivatives,
and so forth. The derivatives are termed the n-th order derivative e.g, second order derivative, third order derivative.
A common alternative notation is,
The dashes are actually roman numerals: the symbol for the forth order derivative would be a slanted IV.
Picking up our earlier example; , the second derivative with respect to is
All subsequent derivatives would be zero. For any polynomial of order n-1, 
,
Other than the second derivative these have no real applications[Citation Needed]. A function which is differentiable infinite times is termed a smooth function.
