11 Apr 2026

Calculus with Maxima

Core Maxima commands for differentiation, integration, limits, series, sums, and differential equations.

maxima calculus

Maxima provides a compact, symbolic workflow for standard calculus operations—differentiation, integration, limits, summation/product manipulation, series expansion, algebraic equation solving, and differential equation solving—supporting both routine computation and physics-oriented analytical work.

Differentiation and Physical Interpretation

Differentiation in Maxima is performed with diff, yielding first or higher derivatives with respect to a chosen variable. In applications, derivatives encode local change, slopes, and time-evolution quantities such as velocity and acceleration, and they appear centrally in higher-order differential models.

• First derivative: diff(f, x); computes $\frac{df}{dx}$.
• $n$th derivative: diff(f, x, n); computes $\frac{d^n f}{dx^n}$.

diff(x^3 + sin(x), x);
diff(exp(x)*cos(x), x, 2);

Defining dependent variables and partial derivatives is also straightforward:

depends(y, x);
diff(y, x);
diff(y, x, 2);

Assumptions of constants and variables

Here is a list of possible assumptions examples:

assume(x > 0);
assume(n, integer);
assume(a, real);

Kill command

The kill command is used to clear variables, functions, or assumptions from the Maxima environment. This is particularly useful for resetting the state of the session or removing specific definitions that are no longer needed.

• Clear a variable: kill(x); removes the definition of variable x.
• Clear a function: kill(f); removes the definition of function f.
• Clear all variables and functions: kill(all); resets the entire environment.

Integration, Limits, and Discrete Operators

Integration via integrate supports both indefinite and definite forms, including improper integrals such as $\int_0^\infty f(x)\,dx$, which commonly arise in normalization and expectation-value calculations. Limits computed by limit are essential for testing continuity, diagnosing singular behavior, and extracting asymptotic structure. Discrete calculus and special-function manipulations frequently use finite sums and products.

• Indefinite integral: integrate(f, x); returns $\int f\,dx$.
• Definite integral: integrate(f, x, a, b); returns $\int_a^b f\,dx$.
• Limit: limit(f, x, a); returns $\lim_{x\to a} f(x)$.
• Summation: sum(f, k, m, n); returns $\sum_{k=m}^{n} f$.
• Product: product(f, k, m, n); returns $\prod_{k=m}^{n} f$.

integrate(x^2, x);
integrate(exp(-x), x, 0, inf);
limit(sin(x)/x, x, 0);
sum(k^2, k, 1, n);
product(k, k, 1, n);

Series Expansion and Algebraic Solving

Local approximation and perturbative analysis often rely on series expansions. Maxima computes Taylor expansions about a point $x=a$ up to a specified order. Algebraic solving with solve provides closed-form roots and symbolic conditions that are routinely used for stationary points and analytical parameter relations.

• Taylor series: taylor(f, x, a, n); expands $f$ about $x=a$ through order $n$.
• Algebraic solve: solve(eq, x); solves an algebraic equation for $x$.

taylor(exp(x), x, 0, 5);
solve(x^2 - 5*x + 6 = 0, x);

Differential Equations and a Minimal Working Session

Differential equations are solved symbolically with ode2, which is particularly useful for first-order models and canonical linear dynamics encountered in equations of motion, growth–decay systems, and wave or dynamical formulations.

• Differential equation solve: ode2(eq, y(x)); solves for $y(x)$ under the given equation.

ode2('diff(y,x) + y = 0, y, x);

A minimal working session that resets the environment and exercises common operations is:

kill(all)$
diff(x^4, x);
integrate(x^2, x);
limit((sin(x))/x, x, 0);
sum(k, k, 1, 5);
taylor(log(1+x), x, 0, 4);
solve(x^2 - 1 = 0, x);