How to Do Calculus with Python ?

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. We will use SymPy library to do calculus with python. SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python.

Installation:

pip install sympy

If we want to write any sympy expression, first we have to declare its symbolic variables. To do this, we can use the following two functions :

• sympy.Symbol(): It is used to declare a single variable by passing the variable as a string into its parameter.
• sympy.symbols(): It is used to declare multivariable by passing the variables as a string into its parameter. All the variables must be separated by a space forming a string.

Differentiation

We can differentiate any sympy expression by using diff(func, var) method. The parameter func denotes the sympy expression to be differentiated and var denotes the variable with respect to which we have to differentiate.

Example 1:

Python3

# Importing library
import sympy as sym

# Declaring variables
x, y, z = sym.symbols('x y z')

# expression of which we have to find derivative
exp = x**3 * y + y**3 + z

# Differentiating exp with respect to x
derivative1_x = sym.diff(exp, x)
print('derivative w.r.t x: ',
      derivative1_x)

# Differentiating exp with respect to y
derivative1_y = sym.diff(exp, y)
print('derivative w.r.t y: ',
      derivative1_y)

Output:

derivative w.r.t x:  3*x**2*y
derivative w.r.t y:  x**3 + 3*y**2

We can also find higher derivatives using the diff(func, var, n) method. Here, the parameter n denotes the nth derivative to be found.

Example 2:

Python3

# Finding second derivative
# of exp with respect to x
derivative2_x = sym.diff(exp, x, 2)
print('second derivative w.r.t. x: ',
      derivative2_x)

# Finding second derivative
# of exp with respect to y
derivative2_y = sym.diff(exp, y, 2)
print('second derivative w.r.t. y: ',
      derivative2_y)

Output:

second derivative w.r.t. x:  6*x*y
second derivative w.r.t. y:  6*y

Integration

You can do indefinite and definite integration of transcendental elementary and special functions via integrate() function. 

Syntax for indefinite integration: sympy.integrate(func, var)

Syntax for definite integration: sympy.integrate(func, (var,  lower_limit, upper_limit))

The parameter func denotes the sympy expression to be differentiated, var denotes the variable with respect to which we have to differentiate, lower_limit denotes to the lower limit of the definite integration and upper_limit denotes the upper limit of the definite integration.

Note: ∞ in SymPy is oo.

Example 1:

Python3

# Indefinite integration of cos(x) w.r.t. dx
integral1 = sym.integrate(sym.cos(x), x)
print('indefinite integral of cos(x): ',
      integral1)

# definite integration of cos(x) w.r.t. dx between -1 to 1
integral2 = sym.integrate(sym.cos(x), (x, -1, 1))
print('definite integral of cos(x) between -1 to 1: ',
      integral2)

# definite integration of exp(-x) w.r.t. dx between 0 to ∞
integral3 = sym.integrate(sym.exp(-x), (x, 0, sym.oo))
print('definite integral of exp(-x) between 0 to ∞: ',
      integral3)

Output:

indefinite integral of cos(x):  sin(x)
definite integral of cos(x) between -1 to 1:  2*sin(1)
definite integral of exp(-x) between 0 to ∞:  1

Limits

You can calculate limit of a function by using limit(function, variable, point). So, if you want to compute the limit of f(x) as  x->0, you would issue limit(f, x, 0).

Example:

Python3

# Calculating limit of f(x) = x as x->∞
limit1 = sym.limit(x, x, sym.oo)
print(limit1)

# Calculating limit of f(x) = 1/x as x->∞
limit2 = sym.limit(1/x, x, sym.oo)
print(limit2)

# Calculating limit of f(x) = sin(x)/x as x->0
limit3 = sym.limit(sym.sin(x)/x, x, 0)
print(limit3)

Output:

oo
0
1

Series Expansion

We can also compute Taylor series expansions of functions around a point. To compute the expansion of f(x) around the point x=x₀ terms of order xⁿ, use sympy.series(f, x, x₀, n). x₀ and n can be omitted, in which case the defaults x₀=0 and n=6 will be used.

Example:

Python3

# assign series
series1 = sym.series(sym.cos(x), x)
print(series1)

# assign series
series2 = sym.series(1/sym.cos(x), x, 0, 4)
print(series2)

Output:

1 - x**2/2 + x**4/24 + O(x**6)
1 + x**2/2 + O(x**4)

The O(x⁴) or O(x⁶) term at the end means that all x terms with a power greater than or equal to x⁴ or x⁶ are omitted.