The derivative

The derivative
Main article: Derivative
Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, we can determine the value of y. This relationship is written as: y = f(x). Where f(x) is the equation for a straight line, y = m x + b, where m and b are real numbers that determine the locus of the line in Cartesian coordinates. m is called the slope and is given by:



where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in". It follows that Δy = m Δx.
In linear functions the derivative of f at the point x is the best possible approximation to the idea of the slope of f at the point x. It is usually denoted f'(x) or dy/dx. Together with the value of f at x, the derivative of f determines the best linear approximation, or linearization, of f near the point x. This latter property is usually taken as the definition of the derivative. Derivatives cannot be calculated in nonlinear functions because they do not have a well-defined slope.
A closely related notion is the differential of a function.



The tangent line at (x, f(x))
When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f' at x. Because the source and target of f are one-dimensional, the derivative of f is a real number. If x and y are vectors, then the best linear approximation to the graph of f depends on how f changes in several directions at once. Taking the best linear approximation in a single direction determines a partial derivative, which is usually denoted ∂y/∂x. The linearization of f in all directions at once is called the total derivative. It is a linear transformation, and it determines the hyperplane that most closely approximates the graph of f. This hyperplane is called the oscilating hyperplane; it is conceptually the same idea as taking tangent lines in all directions at once.

taken from http://en.wikipedia.org/wiki/Differential_calculus