Page 105 - Foundations Of Differential Calculus

88 5. On the Differentiation of Algebraic Functions of One Variable
164. Although the quotient of two functions can be thought of as the
product of two functions, it may be more convenient to use a rule for
differentiating a quotient. Let p/q be a given function whose differential we
1
need to find. When we substitute x + dx for x the quotient becomes
p + dp 1 dq p pdq dp dp dq
=(p + dp) − = − + − .
q + dq q q 2 q q 2 q q 2
When p/q is subtracted, the differential remains,
p dp pdq
d. = − ,
q q q 2
2
since the term dp dq/q vanishes. Hence, we have
p qdp − pdq
d. = ,
q q 2

and the rule for quotients can be stated:
To obtain the differential of a quotient, from the product of the
denominator and the differential of the numerator we subtract
the product of the numerator and the differential of the denom-
inator. Then the remainder is divided by the square of the de-
nominator.

The following examples illustrate the application of this rule.
x
I. If y = , then by this rule
2
a − x 2
2 2 2 2 2
a + x dx − 2x dx a − x dx
dy = 2 = 2 .
2
2
2
2
(a + x ) (a + x )
√
2
a + x 2
II. If y = ,wehave
2
a − x 2

√ √
2 2 2 2 2 2
a − x xdx a + x +2xdx a + x
dy = ,
2
2 2
(a − x )
and when this is reduced we have
2 2
3a + x xdx
dy = 2 √ .
2
2
2
(a − x ) a + x 2
3
2
1 If we wish to keep all terms up to the second order, the term pdq /q cannot be
omitted.