differentiation and derivative Link to heading

  • Differentiation is a technique which can be used for analyzing the way in which a functions change, or in particular, it measures how rapidly a function is changing at any point (HELM, 2008).
  • In mathematics, differentiation is a process of finding the derivative, or rate of change, of a function, which can be carried out by purely algegraic manipulations using three basic derivatives, four rules of operation, and a knowledge of how to manipulate functions (Britannica, 2022).
  • Differentiation is a method of computing a derivative which the rate of change of the output $y$ of the function with respect to the change of the variable $x$ (Nanda, 2023).
  • Differentiation is a process that gives the derivative (TZakrevskiy, 2017).

polynomial function Link to heading

  • It can be represented as its coefficient, e.g.

    $$\tag{1} f(x) = \sum_{i = 0}^n a_i \ x^i \equiv [a_n \ \ a_{n-1} \ \ \dots \ \ a_i \ \ \dots \ \ a_1 \ \ a_0]. $$

  • Its derivative $\displaystyle g(x) = \frac{df(x)}{dx}$ will be

    $$\tag{2} g(x) = \sum_{i = 0}^{n-1} b_i \ x^i \equiv [b_n \ \ b_{n-1} \ \ \dots \ \ b_i \ \ \dots \ \ b_1 \ \ b_0]. $$

  • And the relation between two sets of coefficients $\mathbf{b}$ and $\mathbf{a}$, in term of each element, is

    $$\tag{3} b_i = (i+1) \ a_{i+1}, \ \ \ , i = 1, \ 2, \ 3, \ \dots, \ (n-1). $$

string, inline, function Link to heading

  • A string representation of a polynomial, e.g.
    fstr = 'x^2 + 10x + 1';
  • A function can be constructed from the string, e.g.
    f = inline(fstr);
  • A value can be calculated from the function, e.g.
    f2 = f(2);
  • Complete code is as follow.
    fstr = 'x^2 + 10x + 1';
f = inline(fstr);

fprintf('x\tf(x)\n');
for x = 0:10
  y = f(x);
  fprintf('%.1f\t%.1f\n', x, y);
end
    with
    >> str_inline_func
x	f(x)
0.0	1.0
1.0	12.0
2.0	25.0
3.0	40.0
4.0	57.0
5.0	76.0
6.0	97.0
7.0	120.0
8.0	145.0
9.0	172.0
10.0	201.0
    as the result.

coeffs, string, inline, function Link to heading

  • Previous polynomial function $f(x) = x^2 + 10x + 1$ can definede through its coefficients, e.g.
    b = [1 10 1];
  • String of it can be constructed using poly2sym function, e.g.
    gstr = poly2sym(b);
  • The function is defined through inline function, e.g.
    g = inline(gstr);
  • Value of the function is obtained simply by passing value as argument, e.g.
    g_m1 = g(-1);
  • The complete code is as follow.
    b = [1 10 1];
gstr = poly2sym(b);
g = inline(gstr);
x = -1;
g_m1 = g(x);
fprintf('f(%.2f) = %.2f', x, g_m1);
    with
    >> coeffs_str_inline_func
f(-1.00) = -8.00
    as the result.
  • It is confirmed that $f(-1) = (-1)^2 + 10(-1) + 1 = -8$.

polyder and diff Link to heading

  • Results are as follow.
    >> polyder_diff
function
a = [1, 1, 1, 1, 1]
f(x) = x + x^2 + x^3 + x^4 + 1

derivative using polyder
b = [4, 3, 2, 1]
g(x) = 2*x + 3*x^2 + 4*x^3 + 1

derivative using diff
h(x) = 2*x + 3*x^2 + 4*x^3 + 1
  • And the complete code is given below.
    % polyder_diff

clear;

% define a polynomial function
a = [1 1 1 1 1];
fstr = poly2sym(a);
f = inline(fstr);   % not necessary

% calculate derivative 1st way
b = polyder(a);
gstr = poly2sym(b);
g = inline(gstr);   % not necessary

% calculate derivative 2nd way
hstr = diff(fstr);
h = inline(hstr);   % not necessary

% display initial polynomial function
fprintf('function\n')
fprintf('a = [');
fprintf('%g, ', a(1:end-1));
fprintf('%g]\n', a(end));
fprintf('f(x) = %s\n', fstr);
fprintf('\n');

% display derivative using 1st way
fprintf('derivative using polyder\n')
fprintf('b = [');
fprintf('%g, ', b(1:end-1));
fprintf('%g]\n', b(end));
fprintf('g(x) = %s\n', gstr);
fprintf('\n');

% display derivative using 2nd way
fprintf('derivative using diff\n')
fprintf('h(x) = %s\n', hstr);
fprintf('\n');