program differentiation by tables and with PART FUNCTION

[compsystems]

sorry for my bad English

One way to make a Derivative is through tables rather by a selection of cases, analyzing the parts of the expression

Code:

// version 0.2 Jun 6 2016 by COMPSYSTEMS COPYLEFT inv(©)
#cas
diff_table(xpr,var):=
BEGIN
    LOCAL nparts, operator, part1, part2;
    LOCAL xprSameVar;
    //purge(var);
    //print("");
    //CASE 1: if the expression is a variable name or identifier
    if (type(xpr)==DOM_IDENT) then
        // CASE 2: if the expression is equal to the variable, example diff(x,x)=1, otherwise diff(x,y)=0
        return when(xpr==var,1,0);
    end;
    // number of parts of the expression
    nparts:=part(xpr);
    //operator
    operator:=part(xpr,0);
    //CASE 3: diff(k,v)=0
    //print(nparts); print(operator); wait;
    if (nparts==0) then
        return 0; // xpr=pi, i, numbers
    end;
    if (nparts>1) then
        part1:=part(xpr,1);
        part2:=part(xpr,2);
        //print(part1);print(part2);wait;
        else
        part1:=part(xpr,1);
        //print(part1); wait;
    end;
    
    xprSameVar:= (string(part1)==string(var));
    
    //CASE 4: diff(-f(v),v)=0 // NEG(xpr)
    if (operator=="-") then
        return -1*diff_table(part1,var);
    end;
    //CASE 5: diff(k*f(v),v)=0 // NEG(xpr)
    if (operator="*" and type(part1)==DOM_INT) then
        return part1*diff_table(part2,var);
    end;
    //CASE 6: diff(|f(v)|,v) with f(v)=v
    if (operator=="abs" and xprSameVar) then
        return sign(var); // assuming a function from R -> R
        //return var/abs(var); // Alternate Form
    end;
    //CASE 7: diff(|f(v)|,v)
    if (operator=="abs" and !(xprSameVar)) then
        return sign(var)*diff_table(part1,var);
    end;
    //         //CASE 8: diff(√(f(v)),v)
    //         if operator=="√" and xprSameVar then
    //             return 1/(2*√(var));
    //         end;
    //         //CASE 9: diff(√(v),v)
    //         if operator=="√" and !(xprSameVar) then
    //             return diff_table(part1,var)/(2*√(var));
    //         end;
    
    //CASE 8: diff(ln(v),v)
    if (operator=="ln" and xprSameVar) then
        return 1/var;
    end;
    //CASE 9: diff(ln(f(v)),v)
    if (operator=="ln" and !(xprSameVar)) then
        return diff_table(part1,var)/part1;
    end;
    //fun.trig
    //CASE 10: diff(sin(v),v)
    if (operator="sin" and xprSameVar) then
        return cos(var);
    end;
    //CASE 11: diff(sin(f(v)),v)
    if (operator="sin" and !(xprSameVar)) then
        return cos(part1)*diff_table(p1,var);
    end;
    //CASE 12: diff(cos(v),v)
    if (operator="cos" and xprSameVar) then
        return ­sin(var);
    end;
    //CASE 13: diff(cos(f(v)),v)
    if (operator="cos" and !(xprSameVar)) then
        return ­sin(part1)*diff_table(part1,var);
    end;
    //CASE 14: diff(tan(v),v)
    if (operator="tan" and xprSameVar) then
        return sec(var)^2 ; // alternate form (1/cos(x))^2
    end;
    //CASE 15: diff(tan(f(v)),v)
    if (operator="tan" and !(xprSameVar)) then
        return sec(part1)^2*diff_table(part1,var);
    end;
    //CASE 16: diff(sin^-1(v),v)
    if (operator=="asin" and xprSameVar) then
        return 1/√(1-var^2);
    end;
    //CASE 17: diff(sin^-1(f(v)),v)
    if (operator=="asin" and !(xprSameVar)) then
        return diff_table(part1,var)/√(1-part1^2);
    end;
    //CASE 18: diff(cos^-1(v),v)
    if (operator=="acos" and xprSameVar) then
        return ­1/√(1-var^2);
    end;
    //CASE 19: diff(cos^-1(f(v)),v)
    // acos(-x) -> ((π+2*asin(x))/2)
    //     if (operator=="acos" and !(xprSameVar)) then
    //         return -1*diff_table(part1,var)/√(1-part1^2);
    //     end;
    //CASE 20: diff(atan^-1(v),v)
    if (operator=="atan" and xprSameVar) then
        return 1/(1+var^2);
    end;
    //CASE 21: diff(atan^-1(f(v)),v)
    if (operator=="atan" and !(xprSameVar)) then
        return diff_table(part1,var)/(1+part1^2);
    end;
    
    //CASE  : diff(f+g,v)
    if operator="+" then
        return diff_table(part1,var)+diff_table(part2,var);
    end;
    
    if operator="/" and nparts≠0
  Return (part2*diff_table(part1,var)-part1*diff_table(part2,var))/(part2^2);
    end;

    
    
    // codifying
    // ...
    return Done;
END;
#end

Examples

CASE 1:
diff_table(x,x) -> 1

CASE 2:
diff_table(x,y) -> 0
diff_table(y,x) -> 0

CASE 3:
diff_table(i,x) -> 0
diff_table(√(-1),x) -> 0
diff_table(5,x) -> 0
diff_table(PI,x) -> 0
diff_table(-1,x) -> 0

CASE 4:
diff_table(-x,x) -> -1

CASE 5:
diff_table(-3*x,x) -> -3
diff_table(3*x,x) -> 3
diff_table(+3*x,x) -> 3

CASE 6:
diff_table(abs(x),x) -> sign(x)

CASE 7:
diff_table(abs(-x),x) -> sign(x)

CASE 8:
diff_table(ln(x),x) -> 1/x

CASE 9:
diff_table(ln(-x),x) -> 1/x

CASE 10:
diff_table(sin(x),x) -> cos(x)

CASE 11:
diff_table(sin(-x),x) -> -cos(x)

CASE 12:
diff_table(cos(x),x)­ -> ­-sin(x)

CASE 13:
diff_table(cos(x),x)­ -> ­-sin(x)

CASE 14:
diff_table(tan(x),x)-> (1/cos(x))^2 = sec(x)^2

CASE 15:
diff_table(tan(-x),x)-> -(1/cos(x))^2 = -sec(x)^2

CASE 16:
diff_table(asin(x),x) -> 1/√(1-x^2)

CASE 17:
diff_table(asin(x),x) -> -1/√(1-x^2)

CASE 18:
diff_table(acos(x),x) -> -­1/√(1-x^2)



CASE 20:
diff_table(atan(x),x) -> 1/(1+x^2)

CASE 21:
diff_table(atan(-x),x) -> -1/(1+x^2)

CASE ...:
writing ...


PD: for Bernard How is internally code of differentiation in xcas?