Mathematical Methods for Physics and Engineering - Matematica.NET

2.1 DIFFERENTIATION◮Find the derivative with respect to x of f(t) =2at, wherex = at 2 .We could of course substitute for t and then differentiate f as a function of x, but in thiscase it is quicker to usedfdx = df dtdt dx =2a 12at = 1 t ,where we have used the fact that( ) −1dt dxdx = . ◭dt2.1.4 Differentiation of quotientsApplying (2.6) for the derivative of a product to a function f(x) =u(x)[1/v(x)],we may obtain the derivative of the quotient of two factors. Thus( u) ( ) ′ ′ ( ) )1 1f ′ = = u + u ′ = u(− v′v v v v 2 + u′v ,where (2.12) has been used to evaluate (1/v) ′ . This can now be rearranged intothe more convenient and memorisable form( u) ′f ′ vu ′ − uv ′= =v v 2 . (2.13)This can be expressed in words as the derivative of a quotient is equal to the bottomtimes the derivative of the top minus the top times the derivative of the bottom, allover the bottom squared.◮Find the derivative with respect to x of f(x) =sinx/x.Using (2.13) with u(x) =sinx, v(x) =x and hence u ′ (x) =cosx, v ′ (x) = 1, we findf ′ (x) =x cos x − sin xx 2= cos xx− sin x . ◭x 22.1.5 Implicit differentiationSo far we have only differentiated functions written in the form y = f(x).However, we may not always be presented with a relationship in this simpleform. As an example consider the relation x 3 − 3xy + y 3 = 2. In this case it isnot possible to rearrange the equation to give y as a function of x. Nevertheless,by differentiating term by term with respect to x (implicit differentiation), we canfind the derivative of y.47