Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUSand using (2.6), we obtain, as before omitting the argument,dfdx = u d du(vw) +dx dx vw.Using (2.6) again to expand the first term on the RHS gives the complete resultddw(uvw) =uvdx dx + u dvdx w + du vw (2.9)dxor(uvw) ′ = uvw ′ + uv ′ w + u ′ vw. (2.10)It is readily apparent that this can be extended to products containing any numbern of factors; the expression for the derivative will then consist of n terms withthe prime appearing in successive terms on each of the n factors in turn. This isprobably the easiest way to recall the product rule.2.1.3 The chain ruleProducts are just one type of complicated function that we may encounter indifferentiation. Another is the function of a function, e.g. f(x) =(3+x 2 ) 3 = u(x) 3 ,where u(x) =3+x 2 .If∆f, ∆u and ∆x are small finite quantities, it follows that∆f∆x = ∆f ∆u∆u ∆x ;As the quantities become infinitesimally small we obtaindfdx = df dudu dx . (2.11)This is the chain rule, which we must apply when differentiating a function of afunction.◮Find the derivative with respect to x of f(x) =(3+x 2 ) 3 .Rewriting the function as f(x) =u 3 ,whereu(x) =3+x 2 , and applying (2.11) we finddf du d=3u2dx dx =3u2 dx (3 + x2 )=3u 2 × 2x =6x(3 + x 2 ) 2 . ◭Similarly, the derivative with respect to x of f(x) =1/v(x) may be obtained byrewriting the function as f(x) =v −1 and applying (2.11):df dv= −v−2dx dx = − 1 dvv 2 dx . (2.12)The chain rule is also useful for calculating the derivative of a function f withrespect to x when both x and f are written in terms of a variable (or parameter),say t.46