The symbolic derivative of an expression with respect to a named variable; diff(f, x, n) the n-th derivative (n from 1 to 20). A symbolic call inside is worked out first: diff(diff(f, x), x).
expr (symbolic) -- e.g. x^3wrt (string) -- the variable name, unquoted (e.g. x)n (number) -- optional: the order, a whole number 1 to 20 (default 1)Expands a symbolic expression, distributing products over sums (and over sums inside sums), a power of a product factor by factor. A polynomial comes out highest degree first. Names the script has bound to numbers are put in: with k = 5, expand(k*(x + 1)) is 5x + 5. A power of a power with a whole outer exponent is one power: (x^3)^4 is x^12 and expand((x^2 + 3)^2) is x^4 + 6x^2 + 9; with any other outer exponent it stays as written, in brackets, (x^2)^(1/2).
expr (symbolic) -- e.g. (a+b)^2Simplifies a symbolic expression to canonical form, collecting like terms and like factors.
expr (symbolic) -- Reduces a rational expression in one variable: the common polynomial factor of the top and the bottom is cancelled, and the answer is written as a fraction ((x^2 - 9)/(x + 3) is x - 3; 3x^3/x^5 is 3/x^2; (x^2 - 1)/(x^2 + 2x + 1) is (x - 1)/(x + 1)). Anything else (more than one variable, a function call, a sum of fractions) comes back as collect gives it. The values where a cancelled factor is zero stay undefined in the original expression; the text of a lesson names them.
expr (symbolic) -- e.g. (x^2 - 9)/(x + 3)The symbolic indefinite integral of an expression with respect to a named variable, via a pattern table -- refuses (rather than guesses) outside its covered forms. A power with a letter exponent takes the power rule, the exponent taken as not -1: integrate(V^(-g), V) = V^(1 - g)/(1 - g). An odd power times e^(c x^2) goes by u = x^2: integrate(x^3*exp(-x^2), x) = -(x^2 + 1)e^(-x^2)/2; an even power has no elementary integral (see defint for its moments). The table holds sec, csc, cot, atan, asin, acos, sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh of a linear inside: integrate(sec(x), x) = ln|sec x + tan x|, integrate(csc(x), x) = ln|csc x - cot x|. Powers go by the reduction formulas, roots by the forms of trigonometric substitution, rational functions in exact surds (1/(x^2 - 2) has sqrt(2), never a decimal). A half-angle or tan answer is an antiderivative piece by piece and jumps where tan does; defint will not cross such a jump. A refusal says which of two things it is: "has no closed form in elementary functions" only for what noClosedForm() lists, and otherwise "no method yet", since it may still have one; both name nintegrate for a number over a range. Its search for one question stops at 4000 calls or depth 64, and then says it stopped: counted, never timed, so one input gives one result.
expr (symbolic) -- e.g. x^2wrt (string) -- the variable name, unquoted (e.g. x)Factors a polynomial in one variable over its real (exact) roots: quadratics, rational roots of any degree, common factors and repeated factors. With whole-number coefficients the result is written as a textbook does: whole-number coefficients in each factor, linear factors first, each in descending powers (6x^2 + x - 2 is (3x + 2)(2x - 1)). With no rational root, a factor of degree 2 or 3 with whole coefficients is searched for: x^4 + 4 is (x^2 + 2x + 2)(x^2 - 2x + 2), x^6 + 2x^3 - x^2 + 1 is (x^3 + x + 1)(x^3 - x + 1). In two variables it takes out the common factor, groups terms with a common factor and factors a polynomial whose terms all have one degree (a difference of squares, a trinomial in x and y, a sum or difference of cubes): x^2 - y^2 is (x + y)(x - y). Names bound to numbers are put in first. A polynomial it cannot factor is refused. A whole number is written as its prime factors: factor(38) is 2 * 19, as factorize (any size up to 38 digits, by Miller-Rabin and Pollard's rho; factor(2^64 + 1) is 274177 * 67280421310721). A number of more than 38 digits, or one whose smallest prime factor is above about 10^12, is refused with a note rather than left running. Works up to degree 600.
expr (symbolic) -- e.g. x^2 + 3*x + 2wrt (string) -- optional: the variable name, unquoted (e.g. x); it may be left out when the expression has one variable, and in two variables it is written firstRewrites each logarithm by the product, quotient and power rules, the variables taken as positive: ln(x^2*y/z) is 2 ln x + ln y - ln z, log(b, sqrt(u)) is 1/2 log(b, u), and a power of the base is its exponent (log(2, 8*x) is 3 + log(2, x), log(b, b) is 1).
expr (symbolic) -- an expression with logarithms: ln(u), log(u) (base 10) or log(b, u)Writes the logarithms of a sum as one logarithm for each base, the reverse of expandLog: 2 ln x - ln y + 1/2 ln z is ln(x^2 sqrt(z)/y), a coefficient becoming a power. Terms with no logarithm, and logarithms of other bases, are left beside it.
expr (symbolic) -- an expression with logarithms: ln(u), log(u) (base 10) or log(b, u)Writes each exponential with a number base as a number times a power of x: b^(m x + n) is b^n (b^m)^x, b^m worked out for a whole number m. 2^(x + 3) is 8*2^x, 3^(2x - 1) is 1/3*9^x, 5*2^(3x + 1) is 10*8^x: the initial value and the growth factor read straight off.
expr (symbolic) -- an expression with exponentials of a number base, e.g. 5*2^(3*x + 1)Substitutes named variables into a symbolic expression and evaluates the result numerically, staying exact where the result allows. Where the expression is undefined (a bottom that is zero, as `(x^2 - 9)/(x + 3)` at -3) it is an error, "division by zero", not 0; so is a value outside a function's real domain (`ln(x)` at -1). A call inside it is worked first; if it cannot be, evalAt refuses.
expr (symbolic) -- name (string) -- a variable name, quoted (e.g. "x")value (number) -- the value to substitute for that variableWhether two expressions are the same mathematics, with the page's numbers put in: true when algebra shows it (they expand, reduce or, by the log rules, expand to the same thing) or when they agree at every one of a dozen fixed test points where both are defined (sin(x)^2 + cos(x)^2 and 1); false when a point gives them different values, or when too few points can be evaluated to say. The points are always tried, also after the algebra says equal, so a fault in the algebra shows at a point. In a template's <check> block a false line names the point and both values.
a (symbolic) -- one sideb (symbolic) -- the other sideA product with negative powers written as one fraction: each factor with a negative whole exponent goes under the bar with its exponent made positive, and a fraction in front is split between the two, so `3*x^3*y^-5` shows as 3x^3 over y^5. A power of a product is taken apart first. Names bound to numbers are put in. It is a way of showing the expression; the value to work with further is the expression itself.
expr (symbolic) -- a product of a number and powers