Ecuació de la recta tangent:
u(x) = d_{x}[f(a)]·(x+(-a))+f(a)
Sigui f(x) = x^{2} ==>
u(x) = 2a·(x+(-a))+a^{2}
Sigui f(x) = x ==>
u(x) = x
Ecuació de pla tangent:
u(x,y) = d_{x}[F(a,b)]·(x+(-a))+d_{y}[F(a,b)]·(y+(-b))+F(a,b)
Sigui F(x,y) = x^{2}+y^{2} ==>
u(x,y) = 2a·(x+(-a))+2b·(y+(-b))+( a^{2}+b^{2} )
Sigui F(x,y) = x+y ==>
u(x,y) = x+y
F(x_{j}) és diferenciable <==> F(x_{j}) té hyper-pla tangent <==> ...
... [ED][ D[F(x_{j})] = < d_{x_{1}}[F_{1}(x_{j})],...,d_{x_{n}}[F_{n}(x_{j})] > ].
F(x_{j}) = ...
... int[ d_{x_{1}}[F_{1}(x_{j})] ] d[x_{1}]+...+int[ d_{x_{n}}[F_{n}(x_{j})] ] d[x_{n}]
f(x) = nx^{n+(-1)}+ny^{n+(-1)}·d_{x}[y]
int[ f(x) ]d[x] = x^{n}+y^{n}
y(x) = ( int[ f(x) ]d[x]+(-1)·x^{n} )^{(1/n)}
f(x) = nx^{n+(-1)}+(1/y)·d_{x}[y]
int[ f(x) ]d[x] = x^{n}+ln(y)
y(x) = e^{( int[ f(x) ]d[x]+(-1)·x^{n} )}
f(x) = nx^{n+(-1)}+e^{y}·d_{x}[y]
int[ f(x) ]d[x] = x^{n}+e^{y}
y(x) = ln( int[ f(x) ]d[x]+(-1)·x^{n} )
ecuacions diferencials:
f(x) = nx^{n+(-1)}+mx^{m+(-1)}·d_{x}[y]
int[ f(x) ]d[x] = x^{n}+( x^{m} [o(x)o] y )
y(x) = ( ( int[ f(x) ]d[x]+(-1)·x^{n} )/o(x)o/x^{m} )
f(x) = y^{n}+d_{x}[y]
y(x) = [[(1/(-n)+1)]]( ( (-1)·((-n)+1) )^{(-n)+1}·x , ( int[ f(x) ]d[x] )^{(-n)+1} )
f(x) = e^{y}+d_{x}[y]
y(x) = ln( [[(-1)]]( x , ( e^{int[ f(x) ]d[x]} )^{(-1)} ) )
f(x) = e^{ny}+d_{x}[y]
y(x) = ln( [[(-1)·(1/n)]]( x , ( e^{int[f(x)]d[x]} )^{(-n)} ) )
d_{x}[ h( [[k]](f(x),g(x)) ) ] = ...
... d_{[[k]](f(x),g(x))}[ h( [[k]](f(x),g(x)) ) ]·k·[[k+(-1)]](f(x),g(x))·d_{x}[f(x)]+...
... d_{x}[ h( ( g(x) )^{k} ) ] )
f(x) = ( ln(y) )^{n}+d_{x}[y]
y(x) = ...
.... e^{ [[e[(-1)]-pow[(1/((-n)+1))]]]( (-1)·((-n)+1)·x , ...
... e[1]-pow[(-1)·(1/((-n)+1))]( ln( int[f(x)][x] ) ) ) }
f(x) = ( ln(1/y) )^{n}+d_{x}[y]
y(x) = ...
... e^{ (-1)·[[e[1]-pow[(1/((-n)+1))]]]( ((-n)+1)·x , ...
... e[(-1)]-pow[(-1)·(1/((-n)+1))]( ln( int[f(x)][x] ) ) ) }
càlcul diferencial 2:
ln-pow[n](x) = x^{n}·ln(x) [o] anti-ln-pow[n](x)
e-pow[n](x) = x^{n}·e^{x} [o] anti-e-pow[n](x)
d_{x}[anti-ln-pow[n](x)] = ( anti-ln-pow[n](x)/(nx+( anti-ln-pow[n](x) )^{n}) )
d_{x}[anti-e-pow[n](x)] = (1/x)·( anti-e-pow[n](x)/(n+( anti-ln-pow[n](x) )) )
d_{x}[y] = ( (x^{n+(-1)}y)/(x^{n}+y^{n}) )
y(x) = anti-ln-pow[n]( x^{n}/n )
d_{x}[y] = ( e^{nx}/(e^{nx}+e^{ny}) )
y = ln(u)
d_{x}[u] = ( (e^{nx}u)/(e^{nx}+u^{n}) )
y(x) = ln( anti-ln-pow[n]( e^{nx}/n ) )
d_{x}[y] = ( my/(n+y) )·(1/x)
y(x) = anti-e-pow[n]( x^{m} )
d_{x}[y] = ( my/(n+y) )
y(x) = anti-e-pow[n]( e^{mx} )
F(x,y) = kx+(n+(-k))·y+h( px+qy+(-m) )
0 = k+hp
0 = (n+(-k))+hq
0 = kx+hpx
0 = (n+(-k))y+hqy
0 = kx+(n+(-k))y+hm
h = (-1)·(n/m)
x = 1
y = 1
F(x,y) = kx+(n+(-k))·y+h( px+qy+(-m) )
F(1,1) = n
G(x,y) = kx+(n+(-k))·y+h( px+qy )
G(1,1) = 0
F(x,y) = x^{k}+y^{n+(-k)}+h( px+qy+(-m) )
0 = kx^{k+(-1)}+hp
0 = (n+(-k))·y^{n+(-k)+(-1)}+hq
0 = kx^{k+(-1)}+(n+(-k))y^{n+(-k)+(-1)}+hm
h = (-1)·( (k^{k}+(n+(-k))^{n+(-k)})/m )
x = k
y = (n+(-k))
F(x,y) = x^{k}+y^{n+(-k)}+h( px+qy+(-m) )
F(k,n+(-k)) = k^{k}+(n+(-k))^{n+(-k)}
G(x,y) = x^{k}+y^{n+(-k)}+h( px+qy )
G(k,n+(-k)) = 0
F(x,y) = ke^{x}+(n+(-k))e^{y}+h( px+qy+(-m) )
0 = ke^{x}+hp
0 = (n+(-k))e^{y}+hq
0 = ke^{x}+(n+(-k))e^{y}+hm
h = (-1)·(n/m)
x = 0
y = 0
F(x,y) = ke^{x}+(n+(-k))e^{y}+h( px+qy+(-m) )
F(0,0) = n
G(x,y) = ke^{x}+(n+(-k))e^{y}+h( px+qy )
G(0,0) = 0
F(x,y) = e^{kx}+e^{(2+(-k))y}+h( px+qy+(-m) )
0 = ke^{kx}+hp
0 = (2+(-k))e^{(2+(-k))y}+hq
0 = ke^{kx}+(2+(-k))e^{(2+(-k))y}+hm
h = (-1)·(2/m)
x = 0
y = 0
F(x,y) = e^{kx}+e^{(2+(-k))y}+h( px+qy+(-m) )
F(0,0) = 2
G(x,y) = e^{kx}+e^{(2+(-k))y}+h( px+qy )
G(0,0) = 0
