Re: implicit differentiation

*Subject*: [mg2861] Re: implicit differentiation*From*: wagner at bullwinkle.cs.Colorado.EDU (Dave Wagner)*Date*: 30 Dec 1995 03:51:27 -0600*Approved*: usenet@wri.com*Distribution*: local*Newsgroups*: wri.mathgroup*Organization*: University of Colorado, Boulder*Sender*: daemon at wri.com

In article <4bqlk0$4fd at dragonfly.wri.com>, <marshall at CIS.Edu.HK> wrote: > >How can I do implicit differentiation ? > > > Dt[f,x] gives the derivative of an expression f in which all variables are assumed to depend on x. In := Dt[x^2 Sin[c y], x] Out = 2 x Cos[c y] (y Dt[c, x] + c Dt[y, x]) + 2 x Sin[c y] If no independent variable is specified, the answer is expressed in terms of the total derivatives of each of the unknowns. In := Dt[x^2 Sin[c y]] Out = 2 x Cos[c y] (y Dt[c] + c Dt[y]) + 2 x Dt[x] Sin[c y] There are two ways to specify that a particular variable in a total derivative is a constant. The first way is to used the Constants option to Dt. This is fairly messy. In := Dt[x^2 Sin[c y], Constants->{c}] Out = 2 c x Cos[c y] Dt[y, Constants -> {c}] + 2 x Dt[x, Constants -> {c}] Sin[c y] The second way is to tell Mathematica that a symbol is a constant, permanently. This is done by giving the symbol the attribute "Constant": In := SetAttributes[c, Constant] You can check a symbol's attributes with the Attributes command. In := Attributes[c] Out = {Constant} Now the term Dt[c] goes away, without complicating the result. In := Dt[x^2 Sin[c y]] Out = 2 c x Cos[c y] Dt[y] + 2 x Dt[x] Sin[c y] Here is an implicitly defined function: In := imp[x_,y_] := Log[y] + x y - 1 Solve[imp[x,y]==0, y] Solve::tdep: The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way. Out = Solve[-1 + x y + Log[y] == 0, y] In := Dt[imp[x,y], x] Out = Dt[y, x] y + x Dt[y, x] + -------- y In := Solve[%==0, Dt[y,x]] Out = 2 y {{Dt[y, x] -> -(-------)}} 1 + x y By the way, you can plot implicitly defined functions using the standard package Graphics`ImplicitPlot`. Dave Wagner Principia Consulting (303) 786-8371 dbwagner at princon.com http://www.princon.com/princon