The Derivative
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 is written as y = f(x). If f(x) is the equation for a straight line, then there are two real numbers m and b such that y = m x + b. 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. 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.
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.
Read more about this topic: Differential Calculus
Famous quotes containing the word derivative:
“When we say “science” we can either mean any manipulation of the inventive and organizing power of the human intellect: or we can mean such an extremely different thing as the religion of science the vulgarized derivative from this pure activity manipulated by a sort of priestcraft into a great religious and political weapon.”
—Wyndham Lewis (1882–1957)
“Poor John Field!—I trust he does not read this, unless he will improve by it,—thinking to live by some derivative old-country mode in this primitive new country.... With his horizon all his own, yet he a poor man, born to be poor, with his inherited Irish poverty or poor life, his Adam’s grandmother and boggy ways, not to rise in this world, he nor his posterity, till their wading webbed bog-trotting feet get talaria to their heels.”
—Henry David Thoreau (1817–1862)