Mathematical Methods for Physics and Engineering - Matematica.NET

2.1 DIFFERENTIATIONapproximate the change in the value of the function, ∆f, that results from a smallchange ∆x in x by∆f ≈ df(x) ∆x. (2.2)dxAs one would expect, the approximation improves as the value of ∆x is reduced.In the limit in which the change ∆x becomes infinitesimally small, we denote itby the differential dx, and (2.2) readsdf = df(x) dx. (2.3)dxThis equality relates the infinitesimal change in the function, df, to the infinitesimalchange dx that causes it.So far we have discussed only the first derivative of a function. However, wecan also define the second derivative as the gradient of the gradient of a function.Again we use the definition (2.1) but now with f(x) replaced by f ′ (x). Hence thesecond derivative is defined byf ′′ f ′ (x +∆x) − f ′ (x)(x) ≡ lim, (2.4)∆x→0 ∆xprovided that the limit exists. A physical example of a second derivative is thesecond derivative of the distance travelled by a particle with respect to time. Sincethe first derivative of distance travelled gives the particle’s velocity, the secondderivative gives its acceleration.We can continue in this manner, the nth derivative of the function f(x) beingdefined byf (n) f (n−1) (x +∆x) − f (n−1) (x)(x) ≡ lim. (2.5)∆x→0 ∆xIt should be noted that with this notation f ′ (x) ≡ f (1) (x), f ′′ (x) ≡ f (2) (x), etc., andthat formally f (0) (x) ≡ f(x).All this should be familiar to the reader, though perhaps not with such formaldefinitions. The following example shows the differentiation of f(x) =x 2 from firstprinciples. In practice, however, it is desirable simply to remember the derivativesof standard functions; the techniques given in the remainder of this section canbe applied to find more complicated derivatives.43