Hi Conal, I don't have any pointers, but here's my attempt to derive the general rule for derivatives of application of S: D (S f g) x = d(S f g t) / dt | t = x = d(f t (g t)) / dt | t = x = D1 f t (g t) * dt/dt + D2 f t (g t) * d(g t)/dt | t = x (#) where D1 and D2 are partial derivative wrt to the 1st and 2nd argument, respectively. ( since d(f G H)/dt = D1 f G H * dG/dt + D2 f G H * dH/dt ) We have D1 f t (g t) = lim (e->0) (f (t + e) (g t) - f t (g t))/e = lim (e->0) (C f (g t) (t + e) - C f (g t) t)/e ( since C f x y = f y x ) = D (C f (g t)) t = D ((C f . g) t) t = C D t ((C f . g) t) = S (C D) (C f . g) t and D2 f t (g t) = lim (e->0) (f t (g t + e) - f t (g t))/e = D (f t) (g t) = (D . f) t (g t) = S (D . f) g t Thus D (S f g) x = (#) = S (C D) (C f . g) t + S (D . f) g t * D g t | t = x = S (C D) (C f . g) x + S (D . f) g x * D g x Therefore D (S f g) = S (C D) (C f . g) + S (D . f) g * D g where "+" and "*" are pointwise lifted. Does this look right to you? - Zhanyong -----Original Message----- From: haskell-cafe-admin@haskell.org [mailto:haskell-cafe-admin@haskell.org]On Behalf Of Conal Elliott Sent: Tuesday, May 21, 2002 1:55 PM To: haskell-cafe@haskell.org Subject: derivative (S f g) ? This isn't really a Haskell question, but I'm hoping a fellow Haskeller might have some helpful pointers. Has anyone seen a generalization of the chain rule for derivatives that applies to applications of the S combinator? The conventional chain rule applies to the more restricted composition combinator: D (f . g) = (D f . g) * D g Where D is the differentiation higher-order function, "*" is multiply lifted pointwise to functions (\ a b x -> a x * b x), and "." is function composition. Thanks, - Conal