--[[ Derivative Function with Function Return
     Functional Programming Example -- Higher Order
     H. Conrad Cunningham, Professor
     Computer and Information Science
     University of Mississippi

Developed for CSci 658, Software Language Engineering, Fall 2013

1234567890123456789012345678901234567890123456789012345678901234567890

2013-09-11: Completed prototype

This higher-order differentiation function is adapted from section
1.3.4 of Abelson and Sussman's Structure and Interpretation of
Computer Programs (SICP) textbook.

Note: I leave out the precondition assertion checking to simplify the
presentation.

--]]

-- Function "deriv" takes a single-argument function "f" and a small
-- delta "dx" and returns a single-argument function (closure) that
-- computes an approximation of the value of the derivative at the
-- value of its argument.  
--
-- Definition of derivative: lim(dx->0) (f(x+dx) - f(x)) / dx

local function deriv(f,dx)
  return function(x)  -- return a function (closure)
    return (f(x+dx) - f(x)) / dx
  end
end
 
-- Partial testing

local cube      = function(x) return x * x * x end
local DX        = 0.001
local derivCube = deriv(cube,DX) -- returns one-argument function

local function actualDerivCube(x) return 3 * x * x end

print("\nderivCube(5)             ==> " .. derivCube(5))
print(  "derivative of cube at 5  ==> " .. actualDerivCube(5))

print("\nderiv(cube,0.001)(6)     ==> " .. deriv(cube,0.001)(6))
print(  "derivative of cube at 6  ==> " .. actualDerivCube(6))