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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)

fplot(f)

Figure contains an axes. The axes contains an object of type functionline.

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

f(pi/2)
ans = 

15sym(1/5)

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

fplot(f1) 

Figure contains an axes. The axes contains an object of type functionline.

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

fplot(f2) 

Figure contains an axes. The axes contains an object of type functionline.

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)

fplot(g)

Figure contains an axes. The axes contains an object of type functionline.

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.

subplot(1,2,1) 
fplot(f) 
subplot(1,2,2) 
fplot(g)

Figure contains 2 axes. Axes 1 contains an object of type functionline. Axes 2 contains an object of type functionline.

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)