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Calculus in `SymPy`

Calculus - SymPy 1.12 documentation

Setting Up SymPy​

Before diving into calculus operations, initialize SymPy with symbols and functions:

from sympy import symbols, diff, integrate, limit, series, sin, exp, ln

# Define symbols
x, y, z = symbols('x y z')

Differentiation​

Differentiate functions symbolically:

# Differentiate f(x) = x^2
f_prime = diff(x**2, x)  # Returns 2*x

# Partial differentiation of f(x, y) = x^2 + xy
partial_f_x = diff(x**2 + x*y, x)  # Returns 2*x + y
partial_f_y = diff(x**2 + x*y, y)  # Returns x

Integration​

Perform both definite and indefinite integrations:

# Indefinite integration of f(x) = x^2
integral_f = integrate(x**2, x)  # Returns x**3/3

# Definite integration of f(x) = x^2 from 0 to 1
definite_integral_f = integrate(x**2, (x, 0, 1))  # Returns 1/3

Limits​

Compute the limit of functions as a variable approaches a value:

# Limit of sin(x)/x as x -> 0
lim_sin_x = limit(sin(x)/x, x, 0)  # Returns 1

# Limit of (1+1/x)^x as x -> infinity
lim_exp = limit((1+1/x)**x, x, sympy.oo)  # Returns E

Series Expansion​

Expand functions into their Taylor series around a point:

# Expand exp(x) around 0 up to 5th order
series_exp = series(exp(x), x, 0, 5)  
# Returns 1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5)

# Series expansion of ln(1+x) around 0
series_ln = series(ln(1+x), x, 0, 4)  
# Returns x - x**2/2 + x**3/3 - x**4/4 + O(x**5)

Solving Differential Equations​

SymPy can also solve differential equations symbolically. Here's a basic example:

from sympy import Function, dsolve, Eq, Derivative

# Define an unknown function and differential equation
f = Function('f')
diffeq = Eq(Derivative(f(x), x, x) - 2*Derivative(f(x), x) + f(x), sin(x))

# Solve the differential equation
sol = dsolve(diffeq, f(x))