Learn Calculus in the Live Editor

Learn calculus and applied mathematics using the Symbolic Math Toolbox™. The example shows introductory functions fplot and diff.

To manipulate a symbolic variable, create an object of type syms.

syms x

Once a symbolic variable is defined, you can build and visualize functions with fplot.

f(x) = 1/(5+4*cos(x))
f(x) = 

14cos(x)+51/(4*cos(x) + 5)


Evaluate the function at x=π/2 using math notation.

ans = 


Many functions can work with symbolic variables. For example, diff differentiates a function.

f1 = diff(f) 
f1(x) = 

4sin(x)4cos(x)+52(4*sin(x))/(4*cos(x) + 5)^2


diff can also find the Nth derivative. Here is the second derivative.

f2 = diff(f,2) 
f2(x) = 

4cos(x)4cos(x)+52+32sin(x)24cos(x)+53(4*cos(x))/(4*cos(x) + 5)^2 + (32*sin(x)^2)/(4*cos(x) + 5)^3


int integrates functions of symbolic variables. The following is an attempt to retrieve the original function by integrating the second derivative twice.

g = int(int(f2)) 
g(x) = 

-8tan(x2)2+9-8/(tan(x/2)^2 + 9)


At first glance, the plots for f and g look the same. Look carefully, however, at their formulas and their ranges on the y-axis.


e is the difference between f and g. It has a complicated formula, but its graph looks like a constant.

e = f - g 
e(x) = 

8tan(x2)2+9+14cos(x)+58/(tan(x/2)^2 + 9) + 1/(4*cos(x) + 5)

To show that the difference really is a constant, simplify the equation. This confirms that the difference between them really is a constant.

e = simplify(e) 
e(x) = 1sym(1)