In this page, to deal with the Mathematics function of school textbook.
Mainly, Differential, Integral and Calculation of sum I￡.
c.f. Sample Program: "smp_diff.bas"
Practical usage is described at lower part of the reference.

The formula about 'x' are dealt with by character string.
So formula level operations in string are possible.

3x^2+4x+1a??a?|(1)
3*x^2+4*x+1a??a?|(2)

In mathematics, it is usually written like -(1).
The formula recognized by 'Basic' need '*'(multiplication)
in front of the variable'x' without omission like -(2).

The formula to give to function, it is possible to use both (1) and (2).
((1) is converted to (2) internally, and processed)
The formula(string) returned by function is always returned in format-(2).

The formula(string) returned by format-(2),
it is, -by assigning a numerical value for 'x',
it can calculate by 'calc()' as it is, and able to get the value.

e.g.
x=2 :'substitute 2 for x
print calc(f\$) :'calculate '3*x+5' with assigned value
11 :'result'11' is displayed

And, 'x'-coefficient of formula's calculation result,
it may become irrational number which cannot divide,
then, the result output of formula will be returned by not [a decimal]
but [fraction enclosed by parentheses], like a "(1/3)*x^3+(5/2)*x^2".

e.g. in the case of integral calculation
print intgr\$("x^2+5*x")
(1/3)*x^3+(5/2)*x^2

 GCD

[Features]  To find solution of greatest common divisor.

[Format]  GCD(n1,n2)

[e.g.]
print gcd(30,42)
6

 LCM

[Features]  To find solution of least common multiple.

[Format]  LCM(n1,n2)

[e.g.]
print lcm(30,42)
210

 PRIME

[Features]  To return the first prime number on and after a specified number.

[Format]  PRIME(n)

[Explanation]
When n is a prime number, n is returned,
so it can also use for distinction of whether n is prime number.

[e.g.]
print prime(12)
13

 ROOT

[Features]  To find the n-th root of x.

[Format]  ROOT(n,x)

[Explanation]
This is an approximation.
n√x (n is dimensions number, the upper left mini symbol, not multiplication)
A number that is multiplied by 'n' times to became 'x'.
SQR is only the square, it can find n-th root of three or more dimensions.

[e.g.]
r=root(3,100)
print r
4.6415889136718755
print r^3
100.00000517446294

 FAC

[Features]  The factorial of n is returned.

[Format]  FAC(n)

[Explanation]
(The product of all the integers from 1 to n)
The case of 'fac(5)', it will be 1*2*3*+4*5=120.

[e.g.]
print fac(5)
120

 PERM

[Features]  To return number of cases of permutation of mathematics.

[Format]  PERM(n,r)

[Explanation]
It is calculation represented by 'nPr'.
The total number of branches that take 'r' pieces out
from different 'n' pieces, and put in order.
The case of '6P3', it will be 6*5*4=120.

[e.g.]
print perm(6,3)
120

 COMB

[Features]  To return number of cases of combination of mathematics.

[Format]  COMB(n,r)

[Explanation]
It is calculation represented by 'nCr'.
The total pattern number that select 'r' pieces from different 'n' pieces.
This value is obtained by PERM(n,r)/FAC(r).

[e.g.]
print comb(5,3)
10

 SIGMA

[Features]  The sum of number sequence is calculated.(mathematics Σ)

[Format]  SIGMA(n1,n2)

[Explanation]
4
Σ  (x+1)    (in this case n1=1,n2=4)
x=1
The case 'sigma("x+1",1,4)' then,
to substitute '1 to 4: increasing' for 'formula x',
and the value adding all the results is returned.
This 'formula of Σ' can be calculated including general functions as sin(),cos(),etc.

It also have the function to make various settings.
Form2: sigma("int"|"even"|"odd")
Although increment is usually 1,(default a=sigma("int"))
it can specify the following.
add only when odd : a=sigma("odd")

And although the target variable is 'x' by default,
it can change into any variables by 'a=sigma("v:y")'.
Form2: sigma("v:1chaVariableName")
(to describe 1character Variable-Name next to "v:)
The variable specified here is also applied to target variable of
differential/Integration/fcal function.

[e.g.]
print sigma("2*x^2+1",1,4)
64
a=sigma("v:y")
print sigma("2*y^2+1",1,4)
64
a=sigma("odd"):a=sigma("v:x")
print sigma("2*x^2+1",1,4)
22

 DERIV\$

[Features]  The given formula is differentiated and it is made a Derivative function.

[Format]  DERIV\$(formula-string)

[Explanation]
The result is returned by the formula of a character string.
f'(x)=lim  f(x+h)-f(x)
h→0      /h
formula "x^n" then, Derivative function of result will be "n*x^(n-1)"

[e.g.]
print deriv\$("x^2+2*x+1")
2*x+2

 DIFF

[Features]  To find solution of Differential coefficient.

[Format]  DIFF(formula-string,n)

[Explanation]
The formula is made into Derivative function,
and the value which substituted 'n' for 'x' is acquired.
Differential coefficient will be the slope of a tangent at the time of the formula'x=n'.

[e.g.]
print diff("x^2+2*x+1",4)
10

 INTGR\$

[Features]  The given formula is integrated and it is made a Primitive function.

[Format]  INTGR\$(formula-string)

[Explanation]
∫   f(x) dx    [f(x) is "2*x+2" in example case]
It is returned by the formula of a character string.
Integration is the inverse operation of differentiation.
formula "a*x^n" then, the result formula of integration will be "a/(n+1)*x^(n+1)"
The result is the one without the integral constant 'C'.

[e.g.]
print intgr\$("2*x+2")
x^2+2*x
print intgr\$("x^2+5*x")
(1/3)*x^3+(5/2)*x^2

 DINT

[Features]  To find solution of Definite integral.

[Format]  DINT(formula-string,n1,n2)

[Explanation]
4
∫   f(x) dx     [f(x) is "2*x+2" in example case]
1
The formula is made into Primitive function F(x),
and substitute 'x' for 'n1' and 'n2',
and the value F(n2)-F(n1) is acquired.
When formula is 'x^2', Definite integral result become an area of part
enclosed between x-axis and parabola of formula, range n1<=x<=n2.

[e.g.]
print dint("2*x+2",1,4)
21

 FCAL

[Features]  To calculate formula 1 and 2 for 'x'(default) given by string, and return the result as string.

[Explanation]
To specify "add" or "sub"(subtraction) or "mult"(multiplication) with 3rd parameter.
It is possible to calculate formula whose coefficient is integer.
The formula including fractions are not supported at the moment.

[e.g.]
3*x+5
print fcal("x^2+1","2x-2","mult")
2*x^3-2*x^2+2*x-2