GitHub

Calculus

Giac.jl provides powerful symbolic calculus capabilities including differentiation, integration, limits, and Taylor series expansion.

Setup

using Giac
using Giac.Commands: diff, integrate, limit, series

@giac_var x y

Differentiation

Basic Derivatives

Compute derivatives using the diff command:

diff(x^2, x)
# Output: 2*x

diff(x^3, x)
# Output: 3*x^2

Higher-Order Derivatives

Compute second, third, or higher-order derivatives by specifying the order:

diff(x^4, x, 2)   # Second derivative
# Output: 12*x^2

diff(x^5, x, 3)   # Third derivative
# Output: 60*x^2

Chain Rule

The chain rule is applied automatically:

diff(sin(x^2), x)
# Output: 2*x*cos(x^2)

Product Rule

Derivatives of products are handled automatically:

diff(x * sin(x), x)
# Output: sin(x)+x*cos(x)

Partial Derivatives

For multivariable functions, specify the variable:

@giac_var x y

diff(x^2 * y^3, x)
# Output: 2*x*y^3

diff(x^2 * y^3, y)
# Output: 3*x^2*y^2

Integration

Indefinite Integrals

Compute antiderivatives using the integrate command:

integrate(x^2, x)
# Output: x^3/3

integrate(sin(x), x)
# Output: -cos(x)

integrate(exp(x), x)
# Output: exp(x)

Definite Integrals

Specify bounds for definite integrals:

# ∫₀¹ x² dx = 1/3
integrate(x^2, x, 0, 1)
# Output: 1/3

# ∫₀^π sin(x) dx = 2
integrate(sin(x), x, 0, pi)
# Output: 2

Integration Techniques

GIAC automatically applies various integration techniques:

# Integration by parts
integrate(x * exp(x), x)
# Output: (x-1)*exp(x)

# Partial fractions
integrate(1/(x^2-1), x)
# Uses partial fraction decomposition

# Trigonometric integrals
integrate(sin(x)^2, x)
# Output: x/2-sin(2*x)/4

Limits

Basic Limits

Compute limits using the limit command:

# Classic limit: sin(x)/x as x→0
limit(sin(x)/x, x, 0)
# Output: 1

Limits at Infinity

@giac_var inf  # Create infinity symbol

limit(1/x, x, inf)
# Output: 0

limit((x^2+1)/(2*x^2-3), x, inf)
# Output: 1/2

L'Hôpital's Rule Cases

GIAC automatically handles indeterminate forms:

# 0/0 form
limit((exp(x)-1)/x, x, 0)
# Output: 1

# ∞/∞ form
limit(ln(x)/x, x, inf)
# Output: 0

Taylor Series

Series Expansion

Expand functions as Taylor series using the series command:

# exp(x) around x=0, order 4
series(exp(x), x, 0, 4)
# Output: 1+x+x^2/2+x^3/6+x^4/24+O(x^5)

# sin(x) around x=0, order 5
series(sin(x), x, 0, 5)
# Output: x-x^3/6+x^5/120+O(x^6)

# cos(x) around x=0, order 4
series(cos(x), x, 0, 4)
# Output: 1-x^2/2+x^4/24+O(x^5)

Series Around Other Points

Expand around a point other than zero:

# exp(x) around x=1
series(exp(x), x, 1, 3)
# Expansion around x=1

Advanced Topics

Gradient and Hessian

For multivariable calculus:

using Giac.Commands: gradient, hessian

@giac_var x y

f = x^2 + x*y + y^2

# Gradient: [∂f/∂x, ∂f/∂y]
gradient(f, [x, y])  # ToFix
# Output: [2*x+y, x+2*y]

# Hessian matrix
hessian(f, [x, y])
# Output: [[2, 1], [1, 2]]

Implicit Differentiation

using Giac.Commands: implicitdiff

# For x² + y² = 1, find dy/dx
implicitdiff(x^2 + y^2 - 1, x, y)  # ToFix
# Output: -x/y

Notes

  • All calculus operations work symbolically, not numerically
  • For numerical integration, convert results using to_julia and use Julia's quadrature packages
  • The diff command uses Leibniz notation internally
  • Series expansions include the order term O(x^n) by default