we are said to integrate the function ƒ(x), and F(x) is called the indefinite integral of ƒ(x) with respect to x, and is written
∫ ƒ(x) dx.
7. In the process of § 4 the increment Δy is not in general equal to the product of the increment Δx and the derived Differentials. function ƒ′(x). In general we can write down an equation of the form
Δy = ƒ′(x) Δx + R,
in which R is different from zero when Δx is different from zero; and then we have not only
lim.Δx=0 R = 0,
but also
| lim.Δx=0 | R | = 0. |
| Δx |
We may separate Δy into two parts: the part ƒ′(x)Δx and the part R. The part ƒ′(x)Δx alone is useful for forming the differential coefficient, and it is convenient to give it a name. It is called the differential of ƒ(x), and is written dƒ(x), or dy when y is written for ƒ(x). When this notation is adopted dx is written instead of Δx, and is called the “differential of x,” so that we have
dƒ(x) = ƒ′(x) dx.