Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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)
dx

dx. (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, we

can 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→ 0

f′(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)being

defined by


f(n)(x)≡lim
∆x→ 0

f(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 formal

definitions. 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.

Free download pdf