The derivative and differentiation

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 = ƒ(x), where ƒ 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 slope formula gives an exact value for the slope of a straight line. If the function ƒ is not linear (i.e. its graph is not a straight line), however, when 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.

The idea, is to compute the rate of change as the limiting value of the ratio of the differences Δy / Δx as Δx becomes infinitely small.

Definition via difference quotients

Let ƒ be a real valued function. In classical geometry, the tangent line to the graph of the function ƒ at a real number a was the unique line through the point (a, ƒ(a)) that did not meet the graph of ƒ transversally, meaning that the line did not pass straight through the graph. The derivative of y with respect to x at a is, geometrically, the slope of the tangent line to the graph of ƒ at a. The slope of the tangent line is very close to the slope of the line through (a, ƒ(a)) and a nearby point on the graph, for example (a + h, ƒ(a + h)). These lines are called secant lines. A value of h close to zero gives a good approximation to the slope of the tangent line, and smaller values (in absolute value) of h will, in general, give better approximations. The slope m of the secant line is the difference between the y values of these points divided by the difference between the x values, that is,

This expression is Newton's difference quotient. The derivative is the value of the difference quotient as the secant lines approach the tangent line. Formally, the derivative of the function ƒ at a is the limit