Derivative - Differentiation and The Derivative

Differentiation and The Derivative

Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x. This functional relationship is often denoted y = f(x), where f denotes the function. If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.

The simplest case is when y is a linear function of x, meaning that the graph of y divided by x is a straight line. In this case, y = f(x) = m x + b, for real numbers m and b, and the slope m is given by

where the symbol Δ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in." This formula is true because

y + Δy = f(x+ Δx) = m (x + Δx) + b = m x + b + m Δx = y + mΔx.

It follows that Δy = m Δx.

This gives an exact value for the slope of a straight line. If the function f is not linear (i.e. its graph is not a straight line), however, then the change in y divided by the change in x varies: differentiation is a method to find an exact value for this rate of change at any given value of x.

Figure 2. The secant to curve y= f(x) determined by points (x, f(x)) and (x+h, f(x+h)) Figure 3. The tangent line as limit of secants

The idea, illustrated by Figures 1-3, is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.

In Leibniz's notation, such an infinitesimal change in x is denoted by dx, and the derivative of y with respect to x is written

suggesting the ratio of two infinitesimal quantities. (The above expression is read as "the derivative of y with respect to x", "d y by d x", or "d y over d x". The oral form "d y d x" is often used conversationally, although it may lead to confusion.)

The most common approach to turn this intuitive idea into a precise definition uses limits, but there are other methods, such as non-standard analysis.

Read more about this topic:  Derivative

Other articles related to "differentiation and the derivative, derivative":

Differentiation and The Derivative - Inflection Point
... A point where the second derivative of a function changes sign is called an inflection point ... At an inflection point, the second derivative may be zero, as in the case of the inflection point x=0 of the function y=x3, or it may fail to exist, as in the case of the inflection point x=0 of the ...

Famous quotes containing the word derivative:

    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)