PRELIMINARY CALCULUS
A
P
xf(x)x+∆xf(x+∆x)∆fθ∆xFigure 2.1 The graph of a functionf(x) showing that the gradient or slope
of the function atP,givenbytanθ, is approximately equal to ∆f/∆x.Figure 2.1 shows an example of such a function. Near any particular point,
P, the value of the function changes by an amount ∆f, say, asxchanges
by a small amount ∆x. The slope of the tangent to the graph off(x)atP
is then approximately ∆f/∆x, and the change in the value of the function is
∆f=f(x+∆x)−f(x). In order to calculate the true value of the gradient, or
first derivative, of the function atP, we must let ∆xbecome infinitesimally small.
We therefore define the first derivative off(x)as
f′(x)≡df(x)
dx≡lim
∆x→ 0f(x+∆x)−f(x)
∆x, (2.1)provided that the limit exists. The limit will depend in almost all cases on the
value ofx. If the limit does exist at a pointx=athen the function is said to be
differentiable ata; otherwise it is said to be non-differentiable ata.Theformal
concept of a limit and its existence or non-existence is discussed in chapter 4; for
present purposes we will adopt an intuitive approach.
In the definition (2.1), we allow ∆xto tend to zero from either positive ornegative values and require the same limit to be obtained in both cases. A
function that is differentiable atais necessarily continuous ata(there must be
no jump in the value of the function ata), though the converse is not necessarily
true. This latter assertion is illustrated in figure 2.1: the function is continuous
at the ‘kink’Abut the two limits of the gradient as ∆xtends to zero from
positive or negative values are different and so the function is not differentiable
atA.
It should be clear from the above discussion that near the pointP we may