Calculus

Calculus here is genuinely symbolic: the rules are applied to the expression itself, so the answers are exact rather than numerical approximations.

Derivatives

der(expression, variable) differentiates. derivative is the same function under a longer name.

der(x^3, x) // 3x^2
der(x^3+x, x) // 3x^2+1
der(sin(x), x) // cos(x)
derivative(exp(x), x) // exp(x)

A third argument repeats the differentiation.

der(x^3, x, 2) // 6x
der(x^3, x, 3) // 6

The product, quotient and chain rules all apply, so composed expressions work without anything special.

der(x*y, x) // y

A function whose derivative is not known is left as an unevaluated der call rather than guessed at.

Integrals

integral(expression, variable) finds an indefinite integral. The constant of integration is left off, as is conventional for a calculator.

integral(x^2, x) // 1/3x^3
integral(3x^2+2x+1, x) // x^3+x^2+x
integral(cos(x), x) // sin(x)
integral(1/x, x) // log(x)

Rational functions

Any quotient of polynomials is integrable, and this is the one family where that is a guarantee rather than a table lookup. There is no single rule for a rational function, so it is first split into partial fractions, and each of those pieces does have a rule: a logarithm, a power, or an arctangent.

integral((3x+5)/(x^2-1), x) // 4*log(x-1)-log(x+1)
integral(x^2/(x^2+1), x) // x-atan(x)
integral(1/(x-1)^2, x) // -1/(x-1)
integral(1/(x^2+2x+2), x) // atan(x+1)

The one shape left out is a denominator with a repeated irreducible quadratic factor, such as 1/(x^2+1)^2, which needs a reduction formula rather than the three rules above.

What integration cannot do

Unlike differentiation, integration has no method that always succeeds. Many ordinary-looking expressions have no elementary antiderivative at all, and for those this says so rather than returning something approximate.

integral(exp(x^2), x) // Cannot integrate this: no elementary antiderivative is known for this expression.

That is deliberate. A wrong integral is indistinguishable from a right one wherever it gets used, so reporting the limit is more useful than hiding it.

What is covered: any polynomial, a constant, any rational function whose denominator has no repeated irreducible quadratic factor, the standard functions exp, sin, cos and log applied to a linear argument, sums of any of those, and a constant multiple of any of those.

Taylor series

taylor(expression, variable = point, degree) expands about a point.

taylor(exp(x), x=0, 4) // 1/24x^4+1/6x^3+0.5x^2+x+1
taylor(sin(x), x=0, 5) // 1/120x^5-1/6x^3+x

The coefficients are exact, because each one is a derivative evaluated at the point and then divided by a factorial, all in exact arithmetic. A series whose coefficients would not come out exactly is reported rather than rounded.

Jacobians

jacobian(f1, f2, ...) builds the matrix of partial derivatives, one row per function. The variables are taken from the functions themselves, in alphabetical order.

jacobian(x*y, x+y) // [y, x; 1, 1]