d_{x}[ bab[n][a,b,c](x) ] = ...
... (a·x^{2}+b·x·bab[n][a,b,c](x)+c·( bab[n][a,b,c](x) )^{2})/(x+bab[n][a,b,c](x))^{n}
d_{y}[ aba[n][a,b,c](y) ] = ...
... (aba[n][a,b,c](x)+y)^{n}/(a·( aba[n][a,b,c](x) )^{2}+b·aba[n][a,b,c](x)·y+c·y^{2})
d_{x}[ kak[1][a,b](x) ]^{2} = d_{x}[ bab[2][a^{2},2ab,b^{2}](x) ]
d_{y}[ aka[1][a,b](y) ]^{2} = d_{y}[ aba[2][a^{2},2ab,b^{2}](y) ]
d_{x}[ ka-x[n](x) ] = ( x/(x+ka-x[n](x))^{n} )
d_{y}[ ak-x[n](y) ] = ( (ak-x[n](y)+y)^{n}/(ak-x[n](y)) )
d_{x}[ ka-y[n](x) ] = ( (ka-y[n](x)+x)^{n}/x )
d_{y}[ ak-y[n](y) ] = ( (ak-y[n](y))/(y+ak-y[n](y))^{n} )
d_{x}[y(x)] = ( x/(x+y) )
y(x) = ka-x[1](x)
d_{x}[y(x)] = ( (x+y)/x )
y(x) = ka-y[1](x)
d_{x}[y(x)] = ( x^{2}+xy )
y(x) = ka-x[(-1)](x)
d_{x}[y(x)] = ( 1/(x^{2}+xy) )
y(x) = ka-y[(-1)](x)
d_{x}[y(x)] = ( (y+x)/y )
y(x) = ak-x[1](x)
d_{x}[y(x)] = ( y/(y+x) )
y(x) = ak-y[1](x)
d_{x}[y(x)] = ( 1/(y^{2}+xy) )
y(x) = ak-x[(-1)](x)
d_{x}[y(x)] = ( y^{2}+xy )
y(x) = ak-y[(-1)](x)
d_{x}[ kak[n][a,b](x) ] = ( (ax+b·kak[n](x))/(x+kak[n](x))^{n} )
d_{y}[ aka[n][a,b](y) ] = ( (aka[n](y)+y)^{n}/(a·aka[n](y)+by) )
d_{x}[ kak[n][a,a](x) ] = a·d_{x}[ kak[n][1,1](x) ]
d_{y}[ aka[n][a,a](y) ] = (1/a)·d_{y}[ aka[n][1,1](y) ]
kak[n][1,0](x) = ka-x[n](x)
aka[n][1,0](y) = ak-x[n](y)
kak[n][0,1](y) = ak-y[n](y)
aka[n][0,1](x) = ka-y[n](x)
d_{x}[y(x)] = ( (ax+by)/(x+y)^{n} )
y(x) = kak[n][a,b](x)
d_{x}[y(x)] = ( (x+y)^{n}/(ax+by) )
y(x) = aka[n][b,a](x)
d_{x}[y(x)] = ( (ax+by)·(x+y)^{n} )
y(x) = kak[(-n)][a,b](x)
d_{x}[y(x)] = ( 1/((x+y)^{n}·(ax+by)) )
y(x) = aka[(-n)][b,a](x)
d_{x}[y] = ( (x+(-y))/(x+y) )
y = ux
u+x·d_{x}[u] = (1+(-u))/(1+u)
(1+u)/((-1)u^{2}+(-2)u+1)·d_{x}[u] = (1/x)
(-1)·(1/2)·ln( (-1)u^{2}+(-2)u+1 ) = ln(x)
(-1)u^{2}+(-2)u+1 = (1/x^{2})
(-1)y^{2}+(-2)yx+x^{2} = 1
y^{2}+2yx+(1+(-1)·x^{2}) = 0
y = (1/2)( (-2)x+(8x^{2}+(-4))^{(1/2)} )
y = (-x)+(2x^{2}+(-1))^{(1/2)}
d_{x}[ (-x)+(2x^{2}+(-1))^{(1/2)} ] = (-1)+(2x^{2}+(-1))^{(-1)(1/2)}·(2x)
d_{x}[y] = ( (x+y)/(x+(-z)) )
d_{x}[z] = ( (x+z)/(x+(-y)) )
y = x+(2x^{2}+(-1))^{(1/2)}
z = x+(-1)·(2x^{2}+(-1))^{(1/2)}
d_{x}[y(x)] = ( ((-1)·e^{y})/(e^{x}+e^{y}) )
y = u+x
1+d_{x}[u] = ( ((-1)·e^{u})/(1+e^{u}) )
(-1)·( (1+e^{u})/(1+2·e^{u}) )·d_{x}[u] = 1
( (-1)+(e^{u}/(1+2·e^{u})) )·d_{x}[u] = 1
(-u)+(1/2)·ln(1+2e^{u}) = x
(1/2)·ln(1+2e^{y+(-x)}) = y
1+2e^{y+(-x)} = e^{2y}
e^{y} = (1/2)·( 2e^{(-x)}+( 4e^{(-2)x}+4 )^{(1/2)} )
e^{y} = ( e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )
y = ln( e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )
e^{y}+(-1)·e^{(-x)} = ( e^{(-2)x}+1 )^{(1/2)}
1+2e^{y+(-x)} = e^{2y}
d_{x}[ ln( e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} ) ] = ...
... ( (-1)+(-1)·e^{(-x)}(e^{(-2)x}+1)^{(-1)(1/2)} )/( 1+e^{x}(e^{(-2)x}+1)^{(1/2)} ) = ...
... ( (-1)+(-1)·e^{(-x)}(e^{y}+(-1)e^{(-x)})^{(-1)} )/( 1+e^{x}(e^{y}+(-1)e^{(-x)}) ) = ...
... ( (-1)·e^{y}·(e^{y}+(-1)·e^{(-x)})^{(-1)} )/(e^{x}e^{y}) ) ...
... ( (-1)·e^{y}/(e^{x}e^{2y}+(-1)·e^{y}) ) ...
... ( (-1)·e^{y}/(e^{x}(1+2e^{y+(-x)})+(-1)·e^{y}) )
d_{x}[y(x)] = ( e^{y}/(e^{x}+(-1)·e^{y}) )
y = u+x
1+d_{x}[u] = ( e^{u}/(1+(-1)·e^{u}) )
(-1)·( (1+(-1)·e^{u})/(1+(-2)·e^{u}) )·d_{x}[u] = 1
( (-1)+( ((-1)·e^{u})/(1+(-2)·e^{u}) ) )·d_{x}[u] = 1
(-u)+(1/2)·ln(1+(-2)e^{u}) = x
(1/2)·ln(1+(-2)e^{y+(-x)}) = y
1+(-2)e^{y+(-x)} = e^{2y}
e^{y} = (1/2)·( (-2)·e^{(-x)}+( 4e^{(-2)x}+4 )^{(1/2)} )
e^{y} = ( (-1)·e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )
y = ln( (-1)·e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} )
e^{y}+e^{(-x)} = ( e^{(-2)x}+1 )^{(1/2)}
1+(-2)e^{y+(-x)} = e^{2y}
d_{x}[ ln( (-1)·e^{(-x)}+( e^{(-2)x}+1 )^{(1/2)} ) ] = ...
... ( 1+(-1)·e^{(-x)}(e^{(-2)x}+1)^{(-1)(1/2)} )/( (-1)+e^{x}(e^{(-2)x}+1)^{(1/2)} ) = ...
... ( 1+(-1)·e^{(-x)}(e^{y}+e^{(-x)})^{(-1)} )/( (-1)+e^{x}(e^{y}+e^{(-x)}) ) = ...
... ( e^{y}·(e^{y}+e^{(-x)})^{(-1)} )/(e^{x}e^{y}) ) ...
... ( e^{y}/(e^{x}e^{2y}+e^{y}) ) ...
... ( e^{y}/(e^{x}(1+(-2)e^{y+(-x)})+e^{y}) )
d_{x}[y(x)] = ( e^{y}/(e^{x}+e^{y}) )
y = u+x
1+d_{x}[u] = ( e^{u}/(1+e^{u}) )
(1+e^{u})·d_{x}[u] = (-1)
u+e^{u} = (-x)
e^{y+(-x)} = (-y)
e^{y}·(1/y) = (-1)·e^{x}
e[(-1)](y) = (-1)·e^{x}
y = ln-e[(-1)]( (-1)·e^{x} )
e^{y}·(1/y) = (-1)·e^{x}
e^{y} = y·(-1)·e^{x}
d_{x}[ln-e[(-1)]( (-1)·e^{x} )] = ...
... ( ln-e[(-1)]( (-1)·e^{x} )/( (-1)+ln-e[(-1)]( (-1)·e^{x} ) ) )·( 1/((-1)·e^{x}) )·(-1)·e^{x}
d_{x}[y(x)] = ( ( (-1)·e^{y} )/(e^{x}+(-1)·e^{y}) )
y = u+x
1+d_{x}[u] = ( ( (-1)·e^{u} )/(1+(-1)·e^{u}) )
(1+(-1)·e^{u})·d_{x}[u] = (-1)
u+(-1)·e^{u} = (-x)
(-1)·e^{y+(-x)} = (-y)
e^{y+(-x)} = y
e^{y}·(1/y) = e^{x}
e[(-1)](y) = e^{x}
y = ln-e[(-1)]( e^{x} )
d_{x}[y(x)] = ( e^{x}/(e^{x}+(-1)·e^{y}) )
y = u+x
1+d_{x}[u] = ( 1/(1+(-1)·e^{u}) )
d_{x}[u] = ( e^{u}/(1+(-1)·e^{u}) )
(e^{-u}+(-1))·d_{x}[u] = 1
(-1)·e^{-u}+(-u) = x
(-1)·e^{x+(-y)} = y
(-1)·e^{x} = y·e^{y}
(-1)·e^{x} = e[1](y)
y = ln-e[1]( (-1)·e^{x} )
d_{y}[e[n](y)] = ( n·e[n](y)+y·e[n](y) )·(1/y)
d_{x}[ln-e[n](x)] = ( (ln-e[n](x))/( n+ln-e[n](x) ) )·(1/x)
(-1)·e^{x} = y·e^{y}
(-1)·e^{x}·(1/y) = e^{y}
d_{x}[ln-e[1]((-1)·e^{x})] = ...
... ( 1/( 1+(1/ln-e[1]( (-1)·e^{x} )) ) )·( 1/((-1)·e^{x}) )·(-1)·e^{x} = ...
... ( 1/( (-1)·e^{x}+(-1)·e^{x}·(1/ln-e[1]( (-1)·e^{x} )) ) )·(-1)·e^{x} = ...
... ( 1/( (-1)·e^{x}+e^{y} ) )·(-1)·e^{x}
d_{x}[y(x)] = ( e^{x}/(e^{x}+e^{y}) )
y = u+x
1+d_{x}[u] = ( 1/(1+e^{u}) )
d_{x}[u] = ( ((-1)·e^{u})/(1+e^{u}) )
(e^{-u}+1)·d_{x}[u] = (-1)
(-1)·e^{-u}+u = (-x)
(-1)·e^{x+(-y)} = (-y)
e^{x} = y·e^{y}
e^{x} = e[1](y)
y = ln-e[1]( e^{x} )
d_{x}[y(x)] = ( xy/(x^{2}+y^{2}) )
u+x·d_{x}[u(x)] = ( u/(1+u^{2}) )
( (-1)·(1/u^{3})+(-1)·(1/u) )·d_{x}[u(x)] = (1/x)
(1/2)·(1/u^{2}) = ln(ux)
(1/2)·x^{2} = y^{2}·ln(y)
(1/2)·x^{2} = ln[2](y)
y = e-ln[2]( (1/2)·x^{2} )
d_{x}[y(x)] = ( xy/(x^{2}+(-1)·y^{2}) )
u+x·d_{x}[u(x)] = ( u/(1+(-1)·u^{2}) )
( (1/u^{3})+(-1)·(1/u) )·d_{x}[u(x)] = (1/x)
(-1)·(1/2)·(1/u^{2}) = ln(ux)
(-1)·(1/2)·x^{2} = y^{2}·ln(y)
(-1)·(1/2)·x^{2} = ln[2](y)
y = e-ln[2]( (-1)·(1/2)·x^{2} )
d_{x}[e-ln[2]( (-1)·(1/2)·x^{2} )] = ...
... ( (-1)·x·e-ln[2]( (-1)·(1/2)·x^{2} ) )/( (-1)·x^{2}+( e-ln[2]( (-1)·(1/2)·x^{2} ) )^{2} )
d_{y}[ln[n](y)] = ( n·ln[n](y)+y^{n} )·(1/y)
d_{x}[e-ln[n](x)] = ( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} )
d_{xx}^{2}[e-ln[n](x)] = ( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} )^{2}+...
... (-1)·( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} )^{2}·...
... ( n+n·( e-ln[n](x) )^{n+(-1)}·( ( e-ln[n](x) )/( nx+( e-ln[n](x) )^{n} ) ) )
e-ln[n](x) = ...
... 1+x+( (1+(-1)·n)+(-1)·n )·(x^{2}/2!)+...
