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Symbolic Calculator

- Computer Algebra System for Symbolic Computation of Fractional Calculus

`int_0^oo ` Math Handbook dx =

How to use? There are 5 ways:

  1. Input or click sin(x) , then click the button for derivative, click again for second derivative, click to solve differential equation, click to differentiate solution for double check, click for integration, click to inverse a function, click for definition, click to simplify, click ......
  2. Input command by use of the first function as command, e.g.
    then hit the button or the ENTER key in your keybord.
  3. Input function, e.g.
  4. Input function by use of "," or ";" as separator for multistatements, e.g.
  5. Input question mark ? to show a list of function, i.e.
  6. Input function and question mark ? to show a function source. e.g.
To do
algebra 1st row
convert sin(x) to exp(x)
convert exp(x) to sin(x)
convert sin(x) to sinh(x)
solve equation for x, solve( exp(x)=2 )
calculus: default variable is x 2nd row
convert sin(x) to integral
limit lim( log(x)/x as x->oo )
differentiate `d/dx sin(x)`
integrate ∫ sin(x) dx
infinite integration integrate( exp(-x) as x->oo )
nth derivative formula `d^n/dx^n sin(x)`
semiderivative `d^(0.5)/dx^(0.5) sin(x)`
semiintegrate `d^(-0.5)/dx^(-0.5) sin(x)`
dsolve solve (fractional) differential equation for y,
dsolve( y'=(x-y)! ), dsolve `d^0.5/dx^0.5 y=sin^((-0.5))(x)`
discrete math: default index variable is k 3th row
convert sin(x) to sum
series( sin(x) )
difference Δ`k^2`
Indefinite sum ∑ 1/k^6
partial sum `sum_(k=0)^n` k
partial sum `sum_(k=1)^n` k
infinite sum =sum( x^k/k! as k->oo )
infinite sum `sum_(k=1)^oo x^k/k`
rsolve solve recurrence equation y(x)-y(x-1)=x
definition( sin(x) )
Numeric math 4th row
All Clear input and memory and plot
numeric limit `lim _(x->0) sin(x)/x`
numeric integrate `int _1^2` sin(x) dx
numeric sum `sum _(x=1)^8` x
numeric solve equation nsolve`( x^2-1=0 )`
Taylor series expansion taylor(sin(x))
inverse( sin(x) )
numeric answer
Show function source. sin(x)?
Interactive Plot: mouse wheel to zoom 5th row
Clear input
polar plot polarplot(sin(4*x))
parametric plot parametricplot( x=sin(t) and y=cos(2*t) )
implicit plot `x^2-y^2=2 and x-y=1`
tangent plot tangentplot(sin(x))
secant plot secantplot(sin(x))
plot sin(x) and `x^2`
symbolic answer

The same color buttons are a pair of inverse operators, its result can be checked each other if it returns origial function or not. Usual keywords are lowercase, which are different from uppercase, e.g. sin is different from Sin. Its default variable is small letter x, but its default index variable in discrete math is k.


Add new function f(x) = `x^2`

  • f(x) = x^2

    Add new rule of derivative `d/dx` f(x_) := 2*x

  • d(f(x_),x_) := 2*x

    Add new rule of integral `int` f(x) dx := F(x)

  • integrate(f(x_),x_) := F(x)

    algebra :
    convert sin(x) to exp(x),

  • convert sin(x) to exp
  • toexp( sin(x) )


  • inverse( sin(x) )
    plot a multivalue function
  • inverse( y=sin(x) )

    tangent at x=1

  • tangent( sin(x),x=1 )
    by default x=0
  • tangent( sin(x) )

    solve equation,

  • solve( x^2-5*x-6=0 )
    by default equation = 0
  • solve( x^2-5*x-6 )

    solve inequalities,

  • solve( 2*x-1>0 )
  • solve( 2*x^2-1>0 )

    topoly convert roots to poly(a and b) as holder of polymonial roots,

  • convert (2 and 3) to poly()
  • topoly( 2 and 3 )
    activate poly() to polymonial
  • activate( polyroot(2,3) )

    topoly convert polymonial to poly(a,b,c) as holder of polymonial coefficients,

  • convert `x^2-5*x+6` to poly
  • topoly( `x^2-5*x+6` )
    activate poly() to polymonial
  • activate( poly(6,-5,1) )

    calculus :

  • lim( log(x)/x as x->inf )
    by default x=infinity,
  • lim(log(x)/x)


  • `d/dx sin(x)`
  • d(sin(x))

    Second order derivative

  • `d^2/dx^2 sin(x)`
  • d(sin(x),x,2)

    sin(0.5,x) is inert holder of the 0.5 order derivative `sin^((0.5))(x)`, it can be activated by activate():

  • activate( sin(0.5,x) )
  • `d^(0.5)/dx^(0.5) sin(x)`
  • d(sin(x),x,0.5)


  • `d^(0.5)/dx^(0.5) sin(x)`
  • semid(sin(x))

    Derivative at x=1

  • `d/dx | _(x=1) x^6`
  • d(x^6, x=1)

    Second order derivative at x=1

  • `d^2/dx^2 | _(x=1) x^6`
  • d(x^6, x=1,2)


  • `int` sin(x) dx
  • integrate(sin(x))


  • `d^(-0.5)/dx^(-0.5) sin(x)`
  • semiint(sin(x))

    Definite integration

  • `int_1^2` sin(x) dx
  • integrate(sin(x), x, 1,2)
  • integrate sin(x) as x from 0 to 1

    differential eqaution :
    dsolve differential equation,

  • dsolve( y' = sin(x-y) )
  • dsolve( y(1,x)=cos(x-y) )
  • dsolve( ds(y)=tan(x-y) )

    dsolve fractional differential equation,

  • dsolve( `(d^0.5y)/dx^0.5=sin^((-0.5))(x)` )
    odetest test solution for differential equation,
  • odetest( exp(2x), `dy/dx=2y` )
  • odetest( exp(4x), `(d^0.5y)/dx^0.5=2y` )

    discrete math: : default index variable is k
    Indefinite sum

  • ∑ k
  • sum(k)

    partial sum of 1+2+ .. + k = ∑ x

  • partialsum(k)

    Definite sum of 1+2+ .. +5 = ∑ x

  • sum(x,x,0,5,1)

    Infinite sum of 1/0!+x/1!+ .. +x^k/k! = sum( x^k/k! as k->oo )

  • infsum(x^k/k!,k)

    Taylor series expansion at x=0,

  • taylor( exp(x) as x=0 )
    by default x=0,
  • taylor(exp(x))

    series expand not only to taylor series,

  • series( exp(x) )
    but aslo to other series expansion,
  • series( zeta(2,x) )

    numeric math:
    numeric solve equation,

  • nsolve( x^2-5*x+6=0 )
  • nsolve(x^2-5*x+6)

    numeric integrate, by default x from 0 to 1.

  • nint( x^2-5*x+6,x,0,1 )
  • nint x^2-5*x+6 as x from 0 to 1
  • nint sin(x)

    numeric computation,

  • n( sin(30 degree) )
  • n sin(30 degree)

    Please read its example and manual of symbolic computation Computer Algebra System.


    Computer Algebra System for symbolic computation of any order of fractional derivative. It has three versions:
    1. Phone version: run on any phone online. It does not requires to download anything.
    2. Java version: Java Applet run on phone and tablet that support Java online and off-line. Please contact us if you want it.

    3. PC version: DOS version run on PC. Its old name is SymbMath, you can download it.
    MathHandbook - Computer Algebra System symbolic computation.
    1. Brief
    2. manual
    3. mathHandbook.doc
    4. Developer document
    5. Functions
    6. Examples

    SymbMath - PC DOS version of symbolic computation Computer Algebra System.

    1. table of content
    2. brief
    3. manual
    4. SymbMath.doc
    5. Lists of Review
    6. Review
    7. download

    See Also

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