Definition
The derivative of a function of one variable, f(x), is another function f'(x). Geometrically, the derivative represents the slope of a line tangent to the graph of f at x.
The derivative is one of the fundamental concepts of the calculus, developed around the same time by both Sir Isaac Newton and Gottfried Wilhelm von Leibniz. Leibniz used a different notation to represent the derivative:
"df/dx" which is read as "the derivative of f with respect to x."
The derivative of the derivative of a function is referred to as the second derivative (and this can go on to third, fourth, fifth, etc.).
Functions of more than one variable do not have a derivative, but have partial derivatives with respect to each independent variable. A partial derivative is obtained by treating all other independent variables as constants and then performing normal differentiation.
Derivation of the formula for the derivative of f(x)
(This is not a rigorous proof, but is typical of what you'd see in a Freshman level Calculus class)
The slope of a secent line through two points, (x1,y1) and (x2,y2), on a the graph of f(x) is given by:
f(x2)-f(x1)
-----------
x2-x1
The difference in x between the two points is change in x or more commonly: deltaX. Thus, this formula can be rewritten as:
f(x1+deltaX)-f(x1)
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deltaX
The tangent line is the same as the secent line, except there is no change in x. So if we take the limit of this expression as deltaX approaches 0, then we get a new function of x which represents the slope of a line tangent to f at x. This is the definition of the derivative.