Yet Another Calculus Text, 2007a

32 CHAPTER 1. DERIVATIVES
Thus, for any nonzero infinitesimal dx,
dy v du
dx = dx − u dv
dx ≃
v(v + dv)
This is the quotient rule.
Theorem 1.7.8. If f and g are differentiable, g(x) ≠0,and
v du
dx − u dv
dx
v 2 . (1.7.35)
q(x) = f(x)
g(x) , (1.7.36)
then
q ′ (x) = g(x)f ′ (x) − f(x)g ′ (x)
(g(x)) 2 . (1.7.37)
Oneconsequenceofthequotientruleisthat,sincewealreadyknowhowto
differentiate polynomials, we may now differentiate any rational function easily.
Example 1.7.11. If
then
f(x) = 3x2 − 6x +4
x 2 ,
+1
f ′ (x) = (x2 + 1)(6x − 6) − (3x 2 − 6x + 4)(2x)
(x 2 +1) 2
= 6x3 − 6x 2 +6x − 6 − 6x 3 +12x 2 − 8x
(x 2 +1) 2
= 6x2 − 2x − 6
(x 2 +1) 2 .
Example 1.7.12. We may use either 1.7.30 or 1.7.37 to differentiate
y = 5
x 2 +1 . (1.7.38)
In either case, we obtain
dy
dx = − 5 d
(x 2 +1) 2 dx (x2 +1)=−
10x
(x 2 +1) 2 . (1.7.39)
Exercise 1.7.9.
Exercise 1.7.10.
Find the derivative of
y =
Find the derivative of
14
4x 3 − 3x .
f(x) = 4x3 − 1
x 2 − 5 .