Page 78 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 4] DERIVATIVES 69

y
On the other hand, dy and y are related. In particular, lim ¼ f ðxÞ means that for any "> 0
0
y dy x!0 x
there exists > 0 such that "< <" whenever j xj < . Now dx is an independent variable
x dx
and the axes of x and dx are parallel; therefore, dx may be chosen equal to x.With this choice
" x < y dy <" x

or
dy " x < y < dy þ " x

From this relation we see that dy is an approximation to y in small neighborhoods of x. dy is called
the principal part of y.
dy
The representation of f by has an algebraic suggestiveness that is very appealing and will appear
0
dx
in much of what follows. In fact, this notation was introduced by Leibniz (without the justification
provided by knowledge of the limit idea) and was the primary reason his approach to the calculus, rather
than Newton’s was followed.

THE DIFFERENTIATION OF COMPOSITE FUNCTIONS
Many functions are a composition of simpler ones. For example, if f and g have the rules of
3 3
correspondence u ¼ x and y ¼ sin u, respectively, then y ¼ sin x is the rule for a composite function
F ¼ gð f Þ. The domain of F is that subset of the domain of F whose corresponding range values are in
the domain of g. The rule of composite function differentiation is called the chain rule and is represented
dy dy du
by ¼ ½F ðxÞ¼ g ðuÞf ðxފ.
0
0
0
dx du dx
In the example
3
dy dðsin x Þ 3 2
dx dx ¼ cos x ð3x dxÞ
The importance of the chain rule cannot be too greatly stressed. Its proper application is essential
in the differentiation of functions, and it plays a fundamental role in changing the variable of integration,
as well as in changing variables in mathematical models involving differential equations.

IMPLICIT DIFFERENTIATION
The rule of correspondence for a function may not be explicit. For example, the rule y ¼ f ðxÞ is
2 5
implicit to the equation x þ 4xy þ 7xy þ 8 ¼ 0. Furthermore, there is no reason to believe that this
equation can be solved for y in terms of x. However, assuming a common domain (described by the
independent variable x) the left-hand member of the equation can be construed as a composition of
functions and differentiated accordingly. (The rules of differentiation are listed below for your review.)
In this example, differentiation with respect to x yields

dy dy
5
2x þ 4 y þ 5xy 4 þ 7 y þ x ¼ 0
dx dx
dy
Observe that this equation can be solved for as a function of x and y (but not of x alone).
dx