z = f(x,y) :
z = A x + B y + 
( x, y), ( x, y) ...
= 
, .. 0 
0. , z = f(x,y) . , x y : dz = A x + B y.
:
, , , z/ x = , z/ y = B
z/ x = lim (A x + ( x, 0))/ x = A + lim ( x, 0) / x = A
x 0 x 0
..,
dz = (z/ x) dx + (z/ y) dy (5)
. z = xy3, u = xyz
.
z/ x = f `x(x,y), z/ y = f `y(x,y) , .
, 4
2z/ x2 = (z/ x)/ x, 2z/ xy = (z/ x)/ y,
2z/ y2 = (z/ y)/ y, 2z/xy = (z/ y)/ x
. n . n 1 .
. , , , .
, 2z/xy = 2z/y x.
. 1-, 2- z = x3y + y4
. 2- z = f(x,y) . 1- ( dx, dy -)
d2z =d(dz) =d[(z/x)dx + (z/y) dy] = ( 2z/ x2)dx2 + 2( 2z/xy)dx dy + ( 2z/y2)dy2 (6)
.
z = f(x,y) , t: x = x(t), y = y(t). z = f(x(t), y(t)) t t dz dt ( . ), ..
(7)
. z = y ln x2, x = sin t, y = cos t, dz/ dt =?
z = f(x,y) y = y(x), . dz/dx dz dx
= 
(8)
, z = f(x,y) c y = y(x) | | z.
. z = y x; y = cos x; dz/dx =?
z = f(x,y) x = x(u, v), y = y(u, v), .. , z/u =?, z/v =?, dz du v const
; 
(9)
. z = x2 + 2y2; x = 4u + 3v; y = 2u 5v; z/ d u =?, z/v =?
.
F(x, y) = 0, = (). dy/ dx.
F(x,y) dx
|
|
|
Þ 
(10)
. exp x2 + x y2 = 0; dy/ dx =?. F(x,y) = exp x2 + x y2 ,
F/x = 2x exp x2 + y2 ; F/y = 2 xy; dy/dx = - (2x exp x2 + y2) / 2xy
F(x,y,z) = 0, , z = z(x, y). z/x, z/y.
