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2.3. THE PRODUCT AND QUOTIENT RULES
Combining rules
One of the challenges to learning to apply various derivative shortcut rules correctly and
effectively is recognizing the fundamental structure of a function. For instance, consider
the function given by
f (x) = x sin(x) +
x 2
cos(x) + 2
.
How do we decide which rules to apply? Our first task is to recognize the overall structure
of the given function. Observe that the function f is fundamentally a sum of two slightly
less complicated functions, so we can apply the sum rule 6 and get
f
′ (x) =
d
dx
x sin(x) +
x 2
cos(x) + 2
=
d
dx
[x sin(x)] +
d
dx
x 2
cos(x) + 2
Now, the left-hand term above is a product, so the product rule is needed there, while
the right-hand term is a quotient, so the quotient rule is required. Applying these rules
respectively, we find that
f
′ (x) = (x cos(x) + sin(x)) +
(cos(x) + 2)2x − x 2 (− sin(x))
(cos(x) + 2) 2
= x cos(x) + sin(x) +
2x cos(x) + 4x 2 + x 2 sin(x)
(cos(x) + 2) 2
.
We next consider how the situation changes with the function defined by
s(y) =
y · 7 y
y 2 + 1
.
Overall, s is a quotient of two simpler function, so the quotient rule will be needed. Here,
we execute the quotient rule and use the notation
d
dy to defer the computation of the
derivative of the numerator and derivative of the denominator. Thus,
s
′ (y) =
(y 2 + 1) ·
d
dy [y · 7 y ] − y · 7 y ·
d
dy
y 2 + 1
(y 2 + 1) 2
.
Now, there remain two derivatives to calculate. The first one,
d
dy [y · 7 y ] calls for use of
the product rule, while the second,
d
dy
y 2 + 1
takes only an elementary application of the
6 When taking a derivative that involves the use of multiple derivative rules, it is often helpful to use the
notation d
dx [ ] to wait to apply subsequent rules. This is demonstrated in each of the two examples presented
here.
2.3. THE PRODUCT AND QUOTIENT RULES
Combining rules
One of the challenges to learning to apply various derivative shortcut rules correctly and
effectively is recognizing the fundamental structure of a function. For instance, consider
the function given by
f (x) = x sin(x) +
x 2
cos(x) + 2
.
How do we decide which rules to apply? Our first task is to recognize the overall structure
of the given function. Observe that the function f is fundamentally a sum of two slightly
less complicated functions, so we can apply the sum rule 6 and get
f
′ (x) =
d
dx
x sin(x) +
x 2
cos(x) + 2
=
d
dx
[x sin(x)] +
d
dx
x 2
cos(x) + 2
Now, the left-hand term above is a product, so the product rule is needed there, while
the right-hand term is a quotient, so the quotient rule is required. Applying these rules
respectively, we find that
f
′ (x) = (x cos(x) + sin(x)) +
(cos(x) + 2)2x − x 2 (− sin(x))
(cos(x) + 2) 2
= x cos(x) + sin(x) +
2x cos(x) + 4x 2 + x 2 sin(x)
(cos(x) + 2) 2
.
We next consider how the situation changes with the function defined by
s(y) =
y · 7 y
y 2 + 1
.
Overall, s is a quotient of two simpler function, so the quotient rule will be needed. Here,
we execute the quotient rule and use the notation
d
dy to defer the computation of the
derivative of the numerator and derivative of the denominator. Thus,
s
′ (y) =
(y 2 + 1) ·
d
dy [y · 7 y ] − y · 7 y ·
d
dy
y 2 + 1
(y 2 + 1) 2
.
Now, there remain two derivatives to calculate. The first one,
d
dy [y · 7 y ] calls for use of
the product rule, while the second,
d
dy
y 2 + 1
takes only an elementary application of the
6 When taking a derivative that involves the use of multiple derivative rules, it is often helpful to use the
notation d
dx [ ] to wait to apply subsequent rules. This is demonstrated in each of the two examples presented
here.