diff.haskell

    diff :: Expr -> Expr
    diff e =
      case e of
        (Lit _) -> Lit 0
        (Const _) -> Lit 0
        (Exp Var) -> Exp Var
        (Exp x) -> diff $ Apply (Exp Var) x
        (Ln Var) -> Inv Var
        (Ln x) -> diff $ Apply (Ln Var) x
        (Cos Var) -> Neg $ Sin Var
        (Cos x) -> diff $ Apply (Cos Var) x
        (Sin Var) -> Cos Var
        (Sin x) -> diff $ Apply (Sin Var) x
        (Tan Var) -> Inv (Pow (Cos Var) (Lit 2))
        (Tan x) -> diff $ Apply (Tan Var) x
        (Const _) -> Lit 0
        (Var) -> Lit 1
        (Add f g) -> let f' = diff f
                         g' = diff g
                     in Add (Apply f' Var) (Apply g' Var)
        (Mul f g) -> let f' = diff f
                         g' = diff g
                     in Add (Mul (Apply f' Var) (Apply g Var))
                            (Mul (Apply f Var) (Apply g' Var))
        (Neg f) -> Neg $ diff f
        (Inv f) -> let f' = diff f
                   in Mul f' (Inv (Pow f (Lit 2)))
        (Pow f g) -> diff $ Exp (Mul (Ln (Apply f Var))
                                     (Apply g Var))
        (Sub l r) -> diff $ Add l (Neg r)
        (Div l r) -> diff $ Mul l (Inv r)
        (Sqrt x) -> diff $ Pow x (Inv (Lit 2))
        (Apply f g) -> let f' = diff f
                           g' = diff g
                       in case g' of
                            Var -> f'
                            _ -> Mul (Apply f' (Apply g Var))
                                     (Apply g' Var)