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.
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