2.1 DIFFERENTIATION
approximate the change in the value of the function, ∆f, that results from a small
change ∆xinxby
∆f≈df(x)
dx∆x. (2.2)As one would expect, the approximation improves as the value of ∆xis reduced.
In the limit in which the change ∆xbecomes infinitesimally small, we denote it
by thedifferentialdx, and (2.2) reads
df=df(x)
dxdx. (2.3)Thisequalityrelates the infinitesimal change in the function,df, to the infinitesimal
changedxthat causes it.
So far we have discussed only the first derivative of a function. However, wecan also define thesecond derivativeas the gradient of the gradient of a function.
Again we use the definition (2.1) but now withf(x) replaced byf′(x). Hence the
second derivative is defined by
f′′(x)≡lim
∆x→ 0f′(x+∆x)−f′(x)
∆x, (2.4)provided that the limit exists. A physical example of a second derivative is the
second derivative of the distance travelled by a particle with respect to time. Since
the first derivative of distance travelled gives the particle’s velocity, the second
derivative gives its acceleration.
We can continue in this manner, thenth derivative of the functionf(x)beingdefined by
f(n)(x)≡lim
∆x→ 0f(n−1)(x+∆x)−f(n−1)(x)
∆x. (2.5)
It should be noted that with this notationf′(x)≡f(1)(x),f′′(x)≡f(2)(x), etc., and
that formallyf(0)(x)≡f(x).
All this should be familiar to the reader, though perhaps not with such formaldefinitions. The following example shows the differentiation off(x)=x^2 from first
principles. In practice, however, it is desirable simply to remember the derivatives
of standard functions; the techniques given in the remainder of this section can
be applied to find more complicated derivatives.