Expressions with One Variable

To differentiate a symbolic expression, use the diff command. The following example illustrates how to take a first derivative of a symbolic expression:

syms x
f = sin(x)^2;
diff(f)

ans =
2*cos(x)*sin(x)

Partial Derivatives

For multivariable expressions, you can specify the differentiation variable. If you do not specify any variable, MATLAB^® chooses a default variable by its proximity to the letter x:

syms x y
f = sin(x)^2 + cos(y)^2;
diff(f)

ans =
2*cos(x)*sin(x)

For the complete set of rules MATLAB applies for choosing a default variable, see Find a Default Symbolic Variable.

To differentiate the symbolic expression f with respect to a variable y, enter:

syms x y
f = sin(x)^2 + cos(y)^2;
diff(f, y)

ans =
-2*cos(y)*sin(y)

Second Partial and Mixed Derivatives

To take a second derivative of the symbolic expression f with respect to a variable y, enter:

syms x y
f = sin(x)^2 + cos(y)^2;
diff(f, y, 2)

ans =
2*sin(y)^2 - 2*cos(y)^2

You get the same result by taking derivative twice: diff(diff(f, y)). To take mixed derivatives, use two differentiation commands. For example:

syms x y
f = sin(x)^2 + cos(y)^2;
diff(diff(f, y), x)

ans =
0

Indefinite Integrals of One-Variable Expressions

Suppose you want to integrate a symbolic expression. The first step is to create the symbolic expression:

syms x
f = sin(x)^2;

To find the indefinite integral, enter

int(f)

ans =
x/2 - sin(2*x)/4

Indefinite Integrals of Multivariable Expressions

If the expression depends on multiple symbolic variables, you can designate a variable of integration. If you do not specify any variable, MATLAB chooses a default variable by the proximity to the letter x:

syms x y n
f = x^n + y^n;
int(f)

ans =
x*y^n + (x*x^n)/(n + 1)

For the complete set of rules MATLAB applies for choosing a default variable, see Find a Default Symbolic Variable.

You also can integrate the expression f = x^n + y^n with respect to y

syms x y n
f = x^n + y^n;
int(f, y)

ans =
x^n*y + (y*y^n)/(n + 1)

If the integration variable is n, enter

syms x y n
f = x^n + y^n;
int(f, n)

ans =
x^n/log(x) + y^n/log(y)

Definite Integrals

To find a definite integral, pass the limits of integration as the final two arguments of the int function:

syms x y n
f = x^n + y^n;
int(f, 1, 10)

ans =
piecewise(n == -1, log(10) + 9/y, n ~= -1,...
 (10*10^n - 1)/(n + 1) + 9*y^n)

If MATLAB Cannot Find a Closed Form of an Integral

If the int function cannot compute an integral, it returns an unresolved integral:

syms x
int(sin(sinh(x)))

ans =
int(sin(sinh(x)), x)

Solve Algebraic Equations with One Symbolic Variable

Use the double equal sign (==) to define an equation. Then you can solve the equation by calling the solve function. For example, solve this equation:

syms x
solve(x^3 - 6*x^2 == 6 - 11*x)

ans =
 1
 2
 3

If you do not specify the right side of the equation, solve assumes that it is zero:

syms x
solve(x^3 - 6*x^2 + 11*x - 6)

ans =
 1
 2
 3

Solve Algebraic Equations with Several Symbolic Variables

If an equation contains several symbolic variables, you can specify a variable for which this equation should be solved. For example, solve this multivariable equation with respect to y:

syms x y
solve(6*x^2 - 6*x^2*y + x*y^2 - x*y + y^3 - y^2 == 0, y)

ans =
    1
  2*x
 -3*x

If you do not specify any variable, you get the solution of an equation for the alphabetically closest to x variable. For the complete set of rules MATLAB applies for choosing a default variable see Find a Default Symbolic Variable.

Solve Systems of Algebraic Equations

You also can solve systems of equations. For example:

syms x y z
[x, y, z] = solve(z == 4*x, x == y, z == x^2 + y^2)

x =
 0
 2
 
y =
 0
 2
 
z =
 0
 8

Substitute Symbolic Variables with Numbers

You can substitute a symbolic variable with a numeric value by using the subs function. For example, evaluate the symbolic expression f at the point x = 1/3:

syms x
f = 2*x^2 - 3*x + 1;
subs(f, 1/3)

ans =
2/9

The subs function does not change the original expression f:

f

f =
2*x^2 - 3*x + 1

Substitute in Multivariate Expressions

When your expression contains more than one variable, you can specify the variable for which you want to make the substitution. For example, to substitute the value x = 3 in the symbolic expression

syms x y
f = x^2*y + 5*x*sqrt(y);

enter the command

subs(f, x, 3)

ans =
9*y + 15*y^(1/2)

Substitute One Symbolic Variable for Another

You also can substitute one symbolic variable for another symbolic variable. For example to replace the variable y with the variable x, enter

subs(f, y, x)

ans =
x^3 + 5*x^(3/2)

Substitute a Matrix into a Polynomial

You can also substitute a matrix into a symbolic polynomial with numeric coefficients. There are two ways to substitute a matrix into a polynomial: element by element and according to matrix multiplication rules.

Element-by-Element Substitution.  To substitute a matrix at each element, use the subs command:

syms x
f = x^3 - 15*x^2 - 24*x + 350;
A = [1 2 3; 4 5 6];
subs(f,A)

ans =
[ 312, 250,  170]
[  78, -20, -118]

You can do element-by-element substitution for rectangular or square matrices.

Substitution in a Matrix Sense.  If you want to substitute a matrix into a polynomial using standard matrix multiplication rules, a matrix must be square. For example, you can substitute the magic square A into a polynomial f:

Substitute the Elements of a Symbolic Matrix

To substitute a set of elements in a symbolic matrix, also use the subs command. Suppose you want to replace some of the elements of a symbolic circulant matrix A

syms a b c
A = [a b c; c a b; b c a]

A =
[ a, b, c]
[ c, a, b]
[ b, c, a]

To replace the (2, 1) element of A with beta and the variable b throughout the matrix with variable alpha, enter

alpha = sym('alpha');
beta = sym('beta');
A(2,1) = beta;
A = subs(A,b,alpha)

The result is the matrix:

A =
[     a, alpha,     c]
[  beta,     a, alpha]
[ alpha,     c,     a]

For more information, see Substitute Elements in Symbolic Matrices.

Explicit Function Plot

Create a 2-D line plot by using fplot. Plot the expression $$x^3-6x^2+11x-6$$.

syms x
f = x^3 - 6*x^2 + 11*x - 6;
fplot(f)

Add labels for the x- and y-axes. Generate the title by using texlabel(f). Show the grid by using grid on. For details, see Add Title and Axis Labels to Chart.

xlabel('x')
ylabel('y')
title(texlabel(f))
grid on

Implicit Function Plot

Plot equations and implicit functions using fimplicit.

Plot the equation $$(x^2+y^2{)}^4=(x^2-y^2{)}^2$$ over $$-1<x<1$$.

syms x y
eqn = (x^2 + y^2)^4 == (x^2 - y^2)^2;
fimplicit(eqn, [-1 1])

3-D Plot

Plot 3-D parametric lines by using fplot3.

Plot the parametric line

syms t
fplot3(t^2*sin(10*t), t^2*cos(10*t), t)

Create Surface Plot

Create a 3-D surface by using fsurf.

Plot the paraboloid $$z=x^2+y^2$$.

syms x y
fsurf(x^2 + y^2)