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

Sec. 3] The Derivatives of Functions Not Represented Explicitly_57
or
= _
dy dy'
Tx
Example 1. Find the derivative x , if
y
Solution. We have yx =1+1=^1 ;
hence, x = -7-.
x x *>
x-\- 1
2. The derivatives of functions represented parametrically. If a function
\s related to an argument x by means of a parameter t t
t *-
then
or, in other notation,
Example 2. Find ^, if
dx
I
=
*t
*JL
t^dx'
dt
x a cos t,
y = a sin /
Solution. We find = a sin/ and -r- = acosf. Whence
dt d\
_ _ >
dx a sin /
3. The derivative of an implicit function. If the relationship between x
and y is given in implicit form,
F(x,y) = Q, (I)
then to find the derivative y'x y' in the simplest cases it is sufficient: 1) to
calculate the derivative, with respect to x, of the left side of equation (1),
taking y as a function of x\ 2) to equate this derivative to zero, that is, to put
and 3) to solve the resulting equation for /'.
Example 3. Find the derivative yx
~F(A:,f/) = 0, (2)
if
0. (3)
Solution. Forming the derivative of the left side of (3) and equating it
ito zero, we get
3*' + 3yV -3a (y + xy') = 0,