dualogia, àlgebra, derivades parcials

x+y(x) = f(x)+nx = 0

f(x)-[+]-sum[n](x) = 0

x = anti-f(x)-[+]-sum[n](0)

y(x) = (-1)·anti-f(x)-[+]-sum[n](0)

x·y(x) = x^{k}·f(x) = 1

f(x)-pow[k](x) = 1

x = anti-f(x)-pow[k](1)

y(x) = ( 1/anti-f(x)-pow[k](1) )

x+y(x) = f(x)+nx^{k+1} = 0

x^{k}·( f(x)-pow[(-k)]-[+] sum[n](x) ) = 0

f(x)-[+]-sum[n]-pow[k](x) = 0

x = anti-f(x)-[+]-sum[n]-pow[k](0)

y(x) = (-1)·anti-f(x)-[+]-sum[n]-pow[k](0)

x+y(x) = f(x)+n_{1}x^{k_{1}+1}+...(m)...+n_{m}x^{k_{m}+1} = 0

f(x)-[+]-sum[n_{1}]-pow[k_{1}]-[+]-...(m)...-[+]-sum[n_{m}]-pow[k_{m}](x) = 0

x = anti-f(x)-[+]-sum[n_{1}]-pow[k_{1}]-[+]-...(m)...-[+]-sum[n_{m}]-pow[k_{m}](0)

y(x) = ...

... (-1)·anti-f(x)-[+]-sum[n_{1}]-pow[k_{1}]-[+]-...(m)...-[+]-sum[n_{m}]-pow[k_{m}](0)

sum[n](x) = nx

f(x)-[+]-(sum[n]-[+]-sum[m])(x) = f(x)-[+]-sum[n+m](x)

f(x)+(nx+mx) = f(x)+(n+m)x

f(x)-[+]-(sum[n]-[+]-sum[0])(x) = f(x)-[+]-sum[n](x)

f(x)-[+]-(sum[n]-[+]-sum[(-n)])(x) = f(x)-[+]-sum[0](x) = f(x)

f(x)-[+]-((sum[a]-[+]-sum[b])-[+]-sum[c])(x) = f(x)-[+]-(sum[a]-[+]-(sum[b]-[+]-sum[c]))(x)

f(x)-[+]-(sum[a]-[+]-sum[b])(x) = f(x)-[+]-(sum[b]-[+]-sum[a])(x)

pow[n] = x^{n}

f(x)-(pow[n]-pow[m])(x) = f(x)-pow[n+m](x)

x^{n}·x^{m}·f(x) = x^{n+m}·f(x)

f(x)-(pow[n]-pow[0])(x) = f(x)-pow[n](x)

f(x)-(pow[n]-pow[(-n)])(x) = f(x)-pow[0](x) = f(x)

f(x)-((pow[a]-pow[b])-pow[c])(x) = f(x)-(pow[a]-(pow[b]-pow[c]))(x)

f(x)-(pow[a]-pow[b])(x) = f(x)-(pow[b]-pow[a])(x)

d_{xy}^{2}[ ( f(x)+g(y) )^{n+2} ] = ...

... (n+2)·(n+1)·( f(x)+g(y) )^{n}·d_{x}[f(x)]·d_{y}[g(y)]

d_{yx}^{2}[ ( f(x)+g(y) )^{n+2} ] = ...

... (n+2)·(n+1)·( f(x)+g(y) )^{n}·d_{y}[g(y)]·d_{x}[f(x)]

d_{xy}^{2}[ ( f(x)·g(y) )^{n+2} ] = ...

... d_{x}[ (n+2)·( f(x)·g(y) )^{n+1}·d_{y}[g(y)]·f(x) ] = ...

... d_{x}[g(y)]d_{x}[f(x)]·( ...

... (n+2)·(n+1)·( f(x)·g(y) )^{n+1}+(n+2)·( f(x)·g(y) )^{n+1} ) = ...

... d_{x}[g(y)]d_{x}[f(x)]·(n+2)^{2}·( f(x)·g(y) )^{n+1}