Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 9]_Differentiation of Implicit Functions_205
Sec. 9. Differentiation of Implicit Functions
1. The case of one independent variable. If the equation f(x, y) 0, where
/ (* y) is a differentiate function of the variables x and y, defines y as a
function of x, then the derivative of this implicitly defined function, provided
that f'y (x, y) ?= 0, may be found from the formula
dy =
dx
f'x (**y)
f'y(x,y)'
Higher-order
a)
derivatives are found by successive differentiation of formula
Example -
1. Find
and -~ if
dx dx 2
Solution. Denoting the left-hand side of this equation by f (x, y), we find
the partial derivatives
f' u (x t y)-=3(x
z + y z 2 (
.
) 2y Whence, applying formula (1), we get
To find the second derivative, differentiate with respect to x the first
tive \vhich we have found, taking
deriva-
into consideration the fact that y is a functiun
of x'
dx 2
yJ
dx\ y J y 2
x -~
dx
y x ( ~
J
)
\ y J
y 2
2. The case of several independent variables. Similarly, if the equation
F (x, y, z) 0, where F (x, y, z) is a differentiate function of the variables
x, y and z, defines z as a function of the independent variables x and y and
F z (x t y, z) ^ 0, then the partial derivatives of this implicitly represented
function can, generally speaking,
dK F' z (x, y, z)
be found from the formulas
'
dl J F'g (x, y, z)
Here is another way of finding the derivatives of the function z: different^
ating the equation F (x, y, z)=0, we find
dF _, , dF
J , dF
, rt
Whence it is possible to determine dz, and, therefore,
dz . dz
TT- and 3- .
dx dy
if
'