The derivative at different points of a differentiable function
Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, there is a corresponding value of y. This relationship can be written as y = f(x). If f(x) is the equation for a straight line (called a linear equation), then there are two real numbers m and b such that y = mx + b. In this "slope-intercept form", the term m is called the slope and can be determined from the formula:
where the symbol Δ (the uppercase form of the Greek letter delta) is an abbreviation for "change in". It follows that Δy = m Δx.
A general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a (see figure). This is often denoted f ′(a) in Lagrange's notation or dy/dx|x = a in Leibniz's notation. Since the derivative is the slope of the linear approximation to f at the point a, the derivative (together with the value of f at a) determines the best linear approximation, or linearization, of f near the point a.
If every point a in the domain of f has a derivative, there is a function that sends every point a to the derivative of f at a. For example, if f(x) = x2, then the derivative function f ′(x) = dy/dx = 2x.
A closely related notion is the differential of a function. 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.