Symbolic functions

For how these behave on a calcsheet, including the substitution rules, the arrow spelling and the toolbar, see Symbolic maths. The sixth, integrate, is a formula without limits and a number with them, so its entry is with the other functions: integrate.

simplify

simplify(expression)
expression → simplify
expression →

Simplifies an expression: collects like terms, cancels factors common to a numerator and denominator, applies the trigonometric, exponential and logarithm identities, and keeps roots exact. Names the sheet defines above the block are substituted in first; the result stays an expression while anything in it is unknown, and becomes a value once nothing is.

Parameters

Parameter	Meaning
expression	Any expression. Sheet-defined names in it are substituted: plain numbers as constants, your own functions by their bodies. A name holding units, a matrix or another symbolic result stays as a symbol.

Returns

The simplified expression, or its value once every name in it is defined. No sign is assumed for a name, so $\sqrt{x^2}$ simplifies to $\left|x\right|$, not $x$. The answer is checked numerically against the expression before it is shown; if the check fails, a plainer simplification that mostly collects like terms is shown instead.

Errors

Message	When
simplify expects 1 argument: simplify(expression)	Wrong number of arguments
A definition used here refers to itself, so it cannot be worked out symbolically	A function in the expression refers to itself, or definitions nest more than eight deep

Example

$$\mathrm{simplify}\left(2x+3x\right)=5x$$ $$\mathrm{simplify}\left(\frac{x^2+2x+1}{x+1}\right)=x+1$$ $$\mathrm{simplify}\left({\sin }^{2}x+{\cos }^{2}x\right)=1$$

expand

expand(expression)
expression → expand

Multiplies out brackets and powers of sums, and collects the terms that result. Substitution and evaluation behave as for simplify.

Parameters

Parameter	Meaning
expression	Any expression. Sheet-defined names are substituted as for simplify.

Returns

The expanded expression, or its value once every name in it is defined. The answer is checked numerically against the expression before it is shown; if the check fails, the brackets are multiplied out without collecting terms that mix two names.

Errors

Message	When
expand expects 1 argument: expand(expression)	Wrong number of arguments
A definition used here refers to itself, so it cannot be worked out symbolically	A function in the expression refers to itself, or definitions nest more than eight deep

Example

$$\mathrm{expand}\left({\left(x+1\right)}^3\right)=x^3+3x^2+3x+1$$ $$\mathrm{expand}\left(\left(x+y\right)\left(x-y\right)\right)=x^2-y^2$$

factor

factor(expression)
expression → factor

Factorises a polynomial over the rational numbers: a common factor is taken out, a repeated factor is gathered into a power, and a factor with no real roots is kept as a quadratic. Substitution and evaluation behave as for simplify.

Parameters

Parameter	Meaning
expression	A polynomial. Sheet-defined names are substituted as for simplify.

Returns

The factorised expression, or its value once every name in it is defined. A leading coefficient is taken out in front, so $6x^2-x-2$ gives $6\left(x-\frac{2}{3}\right)\left(x+\frac{1}{2}\right)$. A polynomial with no factors over the rationals, such as $x^2-2$, is returned unfactored, as is anything that is not a polynomial. In more than one variable only a difference of two squares, a perfect square and a numeric common factor are found. A fraction is not factorised; simplify cancels its common factors instead. The answer is checked numerically against the expression before it is shown; if the check fails, the expression is returned as written.

Errors

Message	When
factor expects 1 argument: factor(expression)	Wrong number of arguments
A definition used here refers to itself, so it cannot be worked out symbolically	A function in the expression refers to itself, or definitions nest more than eight deep

Example

$$\mathrm{factor}\left(x^2-4\right)=\left(x-2\right)\left(x+2\right)$$ $$\mathrm{factor}\left(x^4+4\right)=\left(x^2-2x+2\right)\left(x^2+2x+2\right)$$ $$\mathrm{factor}\left(x^3+3x^2+3x+1\right)={\left(x+1\right)}^3$$

derivative

derivative(expression, variable)
expression → derivative, variable

Differentiates expression with respect to variable, applying the chain rule. The variable is shielded from substitution, staying symbolic even when the sheet defines it, and its definition, if any, is applied to the result afterwards.

Parameters

Parameter	Meaning
expression	Any expression. Sheet-defined names are substituted as for simplify, except variable.
variable	A plain name. Shielded from substitution.

Returns

The derivative as an expression, or its value once nothing in it is unknown. Assigned to a function, $g(x):=\mathrm{derivative}(f(x),x)$, the result is a genuinely callable function.

Errors

Message	When
derivative expects 2 arguments: derivative(expression, variable)	Wrong number of arguments
derivative expects a variable name as its last argument, as in derivative(expression, x)	The last argument is not a plain name, or the toolbar’s placeholder is still empty
A definition used here refers to itself, so it cannot be worked out symbolically	A function in the expression refers to itself, or definitions nest more than eight deep

Example

$$\mathrm{derivative}\left(\sin \left(x^2\right),x\right)=2x\cos \left(x^2\right)$$

With $x:=5$ defined above the block, the variable is still differentiated symbolically and the definition applied afterwards:

$$\mathrm{derivative}\left(x^3,x\right)=75$$

solve

solve(equation, variable)
expression → solve, variable

Finds every root of a polynomial equation in variable. The equation may be written as an equation ($x^2=4$) or as an expression taken as equal to zero ($x^2-4$).

Parameters

Parameter	Meaning
equation	A polynomial in variable of degree 1, 2 or 3, with no other unknown left once sheet-defined names are substituted.
variable	A plain name. Shielded from substitution.

Returns

A vector of every root: the real roots in ascending order, followed by any complex roots.

Errors

Message	When
solve expects 2 arguments: solve(equation, variable)	Wrong number of arguments
solve expects a variable name as its last argument, as in solve(expression, x)	The last argument is not a plain name, or the toolbar’s placeholder is still empty
solve has no closed form for this equation in x — to find a root numerically over a bracket, use root(f, a, b)	Not a polynomial of degree 1–3
solve can only work out x when it is the only unknown left	Another unknown remains after substitution
solve found no root for x	The equation has no root
A definition used here refers to itself, so it cannot be worked out symbolically	A function in the equation refers to itself, or definitions nest more than eight deep

Example

$$\mathrm{solve}\left(x^2-4,x\right)=\left[\begin{array}{cc}-2& 2\end{array}\right]$$

$\mathrm{solve}\left(x^2=4,x\right)$ gives the same result.