I think good treatment echoing John Randall is under http://en.wikipedia.org/wiki/Chain_rule#The_fundamental_chain_rule
> From: Tracy Harms <[EMAIL PROTECTED]> > > higher-order derivatives, functionally". In that posting the chain > rule is given by the following expression: > > deriv (f . g) x = deriv f (g x) . deriv g x > To dispel the confusion, the above should be understood as Dx (f o g) = Dgx f o Dx g > From: John Randall <[EMAIL PROTECTED]> > > Dan Bron wrote: > > > Nor is the original math notation itself without flaw. I notice that the > > left-hand side of the equation is written using function composition: > > > > (f o g)' > > > > but the right-hand side is written with nested function calls: > > > > f'(g(x)) > > > > which is inconsistent. I might've written the equation thus: > > > > (f o g)'(x) = (f' o g)(x) * g'(x) > > The prime notation is not really the best mathematical notation for > transparently expressing the chain rule: Leibniz notation is better: > > dy dy du > -- = -- -- > dx du dx > From: Dan Bron <[EMAIL PROTECTED]> > > Oleg wrote: > > Incorrect form: the first "." is composition, the second > > should be multiplication. The composition on the right > > side should be between f' and first g. > > Tracy responded: > > Between what John Randall wrote and the elaboration by the > > author in the original, I think your attempt at correction > > is unsuccessful. > > I think he got it right. Here's another page on the "chain rule" > (composition > rule for derivatives): > > > http://www.math.uncc.edu/~bjwichno/fall2004-math1242-006/Review_Calc_I/lec_composition_rule.htm > > In particular, check out the graphic at the very top: > > http://www.math.uncc.edu/~bjwichno/fall2004-math1242-006/Images/image12.gif > > Transcribing that into ASCII, we get: > > (f o g)'(x) = f'(g(x)) * g'(x) > > Here, I've substituted ' for the mathematical "prime" symbol [1], o for the > mathematical "composition" symbol [2] and * for the mathematical "dot > product" > symbol [3]. > > Incidentally, I think poor ASCII transliteration was what caused this > confusion > in the first place. Conal wanted to use the symbol "." for both function > composition and dot product. Not coincidentally, problems like these are why > J > symbols are all ASCII-based. > > Nor is the original math notation itself without flaw. I notice that the > left-hand side of the equation is written using function composition: > > (f o g)' > > but the right-hand side is written with nested function calls: > > f'(g(x)) > > which is inconsistent. I might've written the equation thus: > > (f o g)'(x) = (f' o g)(x) * g'(x) > > Having done so, it is easy to get a reasonable approximation of the equation > in > J, with the additional benefit that it's executable: > > o =: @: NB. Composition > D =: D.1 NB. Derivative (i.e. prime-symbol) > x =: +/ . * NB. Dot product (even though it looks like a > cross > product, or the variable "x"). > > NB. (f o g)'(x) = (f' o g)(x) * g' (x) > chain =: 2 : '(f o g) D = (f D o g) x (g D)' > > In J4, we could've gotten even closer: > > f =: [. > g =: ]. > NB. (f o g)'(x) = (f' o g)(x) * g' (x) > chain =: (f o g D) = ((f D) o g x (g D)) > > -Dan > > [1] ? aka U+2032 (i.e. u: 16b2032 ) aka "prime" aka "feet" aka "minutes" > http://www.fileformat.info/info/unicode/char/2032/index.htm > > [2] ? aka U+2218 (i.e. u: 16b2218 ) aka "composite function" aka "APL jot" > http://www.fileformat.info/info/unicode/char/2218/index.htm > > [3] ? aka U+22C5 (i.e. u: 16b22C5 ) aka "dot operator" > http://www.fileformat.info/info/unicode/char/22C5/index.htm > > So if my system can compose Unicode properly, and yours can render it, then > the > formula could be written: > > (f ? g)?(x) = f?(g(x)) ? g?(x) > > and, if a future version of J supports Unicode (rather than just ASCII) > names, > we could assign these characters appropriately: > > ? =: @: > ? =: D.1 > ? =: +/ . * > > and come that much closer to having "executable math". But that's a long way > off. As this post proves (probably), Unicode has a way to go before we can > transmit it reliably. But it would make the APL guys happy. ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
