Calculus: Analytic Geometry and Calculus, with Vectors

3.9 Increments and differentials 195

Such numbers dy and dx are called differentials, and some useful observa-

tions can be made. When f and x are such that f'(x) exists, we can let

dx be any number that pleases us and calculate dy, and, provided f'(x) 0

0, we can also let dy be any number we please and calculate dx. Our

interest in differentials can start to develop when we see that, as Figure

3.95 indicates, the point (x + dx, y + dy) must lie on the line tangent

to the graph of f at the point (x,y). This is true because f'(x) is the

slope of the tangent and (3.94) implies that dy/dx is this slope when

dx 0. Since increments Ay and Ax are numbers such that the point

(x + Ox, y + Ay) lies on the graph of f and differentials dx and dy are

numbers such that the point (x + dx, y + dy) lies on the tangent to

the graph at the point (x,y), it is clear that both of the two equalities

dx = Ax and dy = Ay can be true only when the two points (x + Ax,

y + Ay) and (x + dx, y + dy) coincide at a point of intersection of the

graph and the tangent.

Figure 3.95 Figure 3.951

It is particularly easy to produce the differential formula (3.94) when

we use the Leibniz notation for derivatives. The calculation in the left-

hand column

Y = f(x) Y = x2

dydx= f' (x) dydx= 2x

dy = f(x) dx dy = 2x dx

produces the formula whenever f and x are such that f' (x) exists, and the

calculation in the second column shows how things go when f(x) = x2.

This circumstance emphasizes the fact that, when f and x are such that

f'(x) exists and dz is as usual the derivative f'(x), the differentials dy

and dx are defined in such a way that the quotient dx (dy divided by dx)

is the same as the derivative dx when dx 0 0. To find the differential

formula relating dy and dx when f and x are given, it is therefore sufficient

to set y = f (x), differentiate to obtain the formula

(3.952) dx = f'(x),