If y/f (x ), the notation dy /dx signifies the derivative of f(x) with respect to x so that


dx f ?(x):

Although suggestive, the notation does not mean that f ?(x ) is the quotient of two finite quantities dy and dx . However, it is often helpful to treat dy /dx as

though this were the case, and to do this we introduce small finite numbers

dx and dy called differentials. This is accomplished by considering a differ-

entiable function y/f(x ) defined for some aB/xB/b, taking a fixed point x0 in this interval, defining the differential of x , written dx , to be any small non-

zero number, and the differential of y corresponding to x/x0, written dy, to be given by

dy f ?(x0)dx:

The geometrical meaning of differentials can be seen by examination of

Fig. 55. The tangent line PT to the curve at P (the point (x0, f (x0)) has

gradient tan a/f ?(x0), so the ratio dy : dx of the differentials dx and dy at P equals the gradient of the curve at P. If a small non-zero change in x from x0 is denoted by Dx , we can set dx/Dx , and then the approximate change in f(x ) due to the change Dx in x may be taken to be dy. The exact change Dy will differ from dy, as seen in Fig. 55, but when the differential dx is small,

the differential dy will be a good approximation to the exact change Dy. This is called the tangent line approximation to the graph of y/f(x) at

x/x0. We see from this that

lim Dx00

Dy dy

with dy/f ?(x0) dx and Dy/f (x0/Dx )/f (x0). If the point x0 is allowed to vary, so the suffix zero can be omitted, and we

obtain the expression

dy f ?(x)dx

Example 18.1

Find the differential dy for the function y/f(x) given that (i) f (x )/3x2/4x/3; (ii) f (x )/x sin 2x ; (iii) f (x )/(1/x2)1/2.