## Calculating Derivatives

### May 5, 2017

This is easy in a language that supports higher-order functions, and either difficult or impossible in one that doesn’t. Here’s a Python solution:

Python 2.7.10 (default, Jun 1 2015, 18:05:38) [GCC 4.9.2] on cygwin Type "help", "copyright", "credits" or "license" for more information. >>> def deriv(f): ... dx = 0.0001 ... return lambda x : (f(x+dx) - f(x)) / dx ... >>> def cube(x): ... return x*x*x ... >>> cube_deriv = deriv(cube) >>> print(cube_deriv(2)) 12.00060001 >>> print(cube_deriv(3)) 27.0009000101 >>> print(cube_deriv(4)) 48.0012000099

You can run the program in Scheme or Python.

The real point of this exercise is to get you to look at Joe Marshall’s paean to Scheme. If your language doesn’t make this as easy as Scheme, maybe you need a new language.

Using C11 and function pointers:

#include <stdio.h>

#include <stdlib.h>

double deriv(double (*f)(double), double x);

double cube(double x);

int main(int argc, char **argv) {

printf("%f\n", deriv(cube, 2));

printf("%f\n", deriv(cube, 3));

printf("%f\n", deriv(cube, 4));

exit(0);

}

double deriv(double (*f)(double), double x) {

const double dx = 0.0000001;

const double dy = f(x + dx) - f(x);

return dy / dx;

}

`double cube(double x) {`

return x * x * x;

}

Using a slightly different formula.

Results using the same example as Rutger:

Haskell

Gforth

Prints

Here’s another Haskell version, similar to Josef’s. This example shows that you can use it with various numeric types. For example, using the “constructive reals” (CReal) from the “numbers” package we can specify an arbitrary precision. (Also, note that ^3 is shorthand for the cube function, \x -> x^3.)

MUMPS V1:

EXTR ; Test of calculating derivatives

N I

F I=0:1:9 W !,I,” –> “,$$DERIV(“CUBE”,I,.0000001)

Q

;

CUBE (N) ; Cube a number

Q N*N*N

;

DERIV (FUNC,VAL,DX) ;

Q @(“$$”_FUNC_”(VAL+DX)”)-@(“$$”_FUNC_”(VAL)”)/DX

MCL> D ^EXTR

0 –> 0

1 –> 3.0000003

2 –> 12.0000006

3 –> 27.0000009

4 –> 48.0000012

5 –> 75.0000015

6 –> 108.0000018

7 –> 147.0000021

8 –> 192.0000024

9 –> 243.0000027