5.4 Rate of Change of a Rational Function—The Quotient Rule

CHECK, CONSOLIDATE, COMMUNICATE
1. Use an example to show that the derivative of a rational function is not
the same as the quotient of the derivatives of its numerator and
denominator.
2. Compare the quotient rule to the product rule. What is similar about the
two rules? What is different?
3. Why might you need to find the first and second derivatives of a rational
function? Give an example of a rational function. Then find the first and
second derivatives.
KEY IDEAS
• The derivative of a quotient of two differentiable functions is not the
quotient of their derivatives.
• The quotient rule is a rule for finding the derivative of a rational
function.
f (x)
Let h(x) g(x)
. If both f ′(x) and g′(x) exist, the derivative of h(x) is
h′(x) where g(x) ≠ 0.
• The quotient rule in Leibniz notation is
, g(x) ≠ 0.
d
d
x
f ′(x)g(x) g′(x)f (x)
[g(x)]
2
d
f
dx
(x) g(x) d
f (x)
d
g(x) x
f (x)
g(x) [g(x)]
2
5.4 Exercises
A
1. Find the derivative of each rational function from first principles.
x
(a) f (x) 3
(b) g(x) x
2 (c) h(x)
x
1 x
2. Use the quotient rule to find f ′(x) for each function.
(a) f (x) x
3
x 3
(b) f (x)
(c) f (x)
(d) f (x)
(e) f (x)
x 3 2x
x 2 x 1
(f) f (x)
3x 4
x 2 6
(x 4)(x 5)
2x(x 3)
x
3
1 x
ax b
cx d
3x 2 2
x
(g) f (x)
(i) f (x)
x 2 1
2x 3
1 x 4
x
2
(h) f (x)
5x 4 9
x 2
(j) f (x) 5 1 x
x 3
5.4 RATE OF CHANGE OF A RATIONAL FUNCTION—THE QUOTIENT RULE 383