Differential or Derivatives in MATLAB

Differentiation of a function y = f(x) tells us how the value of y changes with respect to change in x. It can also be termed as the slope of a function.

Derivative of a function f(x) wrt to x is represented as  {\displaystyle f'(x)= \frac {dy}{dx}}

MATLAB allows users to calculate the derivative of a function using diff() method. Different syntax of diff() method are:

• f' = diff(f)
• f' = diff(f, a)
• f' = diff(f, b, 2)

f' = diff(f)

It returns the derivative of function f(x) wrt variable x.

Example 1:

% Create a symbolic expression in variable x
syms x
f = cos(x);
disp("f(x) :");
disp(f);

% Derivative of f(x)
d = diff(f);
disp("Derivative of f(x) :");
disp(d);

Output :

Example 2: Evaluating the derivative of a function at a specified value using subs(y,x,k).

• subs(y,x,k), it gives the value of function y at x = k.

% Create a symbolic expression in 
# variable x
syms x
f = cos(x);
disp("f(x) :");
disp(f);

% Derivative of f(x)
d = diff(f);
val = subs(d,x,pi/2);

disp("Value of f'(x) at x = pi/2:");
disp(val);

Output :

f' = diff(f, a)

• It returns the derivative of function f with respect to variable a.

% Create a symbolic expression in variable x
syms x t;
f = sin(x*t);
disp("f(x) :");
disp(f);

% Derivative of f(x,t) wrt t
d = diff(f,t);
disp("Derivative of f(x,t) wrt t:");
disp(d);

Output :

f' = diff(f, b, 2)

It returns the double derivative of function f with respect to variable b.

Example 1:

% Create a symbolic expression in 
% variable x,n
syms x n;
f = x^n;
disp("f(x,n) :");
disp(f);

% Double Derivative of f(x,n) wrt x
d = diff(f,x,2);
disp("Double Derivative of f(x,n) wrt x:");
disp(d);

Output :

In the same way, you can also calculate the k-order derivative of function f using diff(f,x,k).

Example 2: 

Calculating the partial derivative  {\displaystyle {\frac {\partial (f,g)}{\partial (u,v)}}}     } using Jacobian matrix and determinant.

• {\frac {\partial (f,g)}{\partial (u,v)}} =       {\displaystyle {\begin{aligned}{\begin{vmatrix}{\frac {\partial (f)}{\partial (u)}}&{\frac {\partial (f)}{\partial (v)}}\\\\{\frac {\partial (g)}{\partial (u)}}&{\frac {\partial (g)}{\partial (v)}}\end{vmatrix}}\end{aligned}}}

% Create a symbolic expression in variable 
% u and v
syms u v;
f = u^2;
g = sin(v)*(3*u);
disp("f(u,v) :");
disp(f);
disp("g(u,v) :");
disp(g);

% Jacobian matrix of function f(u,v) and 
% g(u,v)
J = jacobian([f; g], [u v]);
disp("Jacobian matrix :");
disp(J);

% Determinant of Jacobian matrix
d = det(J);
disp("Determinant of Jacobian matrix:");
disp(d);

Output :