2.3. THE PRODUCT AND QUOTIENT RULES
105
Next, we expand our perspective from the specific example above to the more general
and abstract setting of a product p of two differentiable functions, f and g. If we have
P(x) = f (x) · g(x), our work above suggests that P ′ (x) = f (x)g ′ (x) + g(x) f ′ (x). Indeed,
a formal proof using the limit definition of the derivative can be given to show that the
following rule, called the product rule, holds in general.
Product Rule: If f and g are differentiable functions, then their product P(x) =
f (x) · g(x) is also a differentiable function, and
P
′ (x) = f (x)g
′ (x) + g(x) f
′ (x).
In light of the earlier example involving shares of stock, the product rule also makes
sense intuitively: the rate of change of P should take into account both how fast f and g
are changing, as well as how large f and g are at the point of interest. Furthermore, we
note in words what the product rule says: if P is the product of two functions f (the first
function) and g (the second), then “the derivative of P is the first times the derivative of
the second, plus the second times the derivative of the first.” It is often a helpful mental
exercise to say this phrasing aloud when executing the product rule.
For example, if P(z) = z 3 · cos(z), we can now use the product rule to differentiate P.
The first function is z 3 and the second function is cos(z). By the product rule, P ′ will be
given by the first, z 3 , times the derivative of the second, − sin(z), plus the second, cos(z),
times the derivative of the first, 3z 2 . That is,
P
′ (z) = z
3 (− sin(z)) + cos(z)3z
2 = −z
3 sin(z) + 3z
2 cos(z).
The following activity further explores the use of the product rule.
Activity 2.7.
Use the product rule to answer each of the questions below. Throughout, be sure to
carefully label any derivative you find by name. It is not necessary to algebraically
simplify any of the derivatives you compute.
(a) Let m(w) = 3w 17 4 w . Find m ′ (w).
(b) Let h(t) = (sin(t) + cos(t))t 4 . Find h ′ (t).
(c) Determine the slope of the tangent line to the curve y = f (x) at the point
where a = 1 if f is given by the rule f (x) = e x sin(x).
(d) Find the tangent line approximation L(x) to the function y = g(x) at the point
where a = −1 if g is given by the rule g(x) = (x 2 + x)2 x .
⊳
change, while our (informal) discussion has also thought of this number as the total change in the number of
shares over the course of a single day. The formal proof of the product rule reconciles this issue by taking the
limit as the change in the input tends to zero.
105
Next, we expand our perspective from the specific example above to the more general
and abstract setting of a product p of two differentiable functions, f and g. If we have
P(x) = f (x) · g(x), our work above suggests that P ′ (x) = f (x)g ′ (x) + g(x) f ′ (x). Indeed,
a formal proof using the limit definition of the derivative can be given to show that the
following rule, called the product rule, holds in general.
Product Rule: If f and g are differentiable functions, then their product P(x) =
f (x) · g(x) is also a differentiable function, and
P
′ (x) = f (x)g
′ (x) + g(x) f
′ (x).
In light of the earlier example involving shares of stock, the product rule also makes
sense intuitively: the rate of change of P should take into account both how fast f and g
are changing, as well as how large f and g are at the point of interest. Furthermore, we
note in words what the product rule says: if P is the product of two functions f (the first
function) and g (the second), then “the derivative of P is the first times the derivative of
the second, plus the second times the derivative of the first.” It is often a helpful mental
exercise to say this phrasing aloud when executing the product rule.
For example, if P(z) = z 3 · cos(z), we can now use the product rule to differentiate P.
The first function is z 3 and the second function is cos(z). By the product rule, P ′ will be
given by the first, z 3 , times the derivative of the second, − sin(z), plus the second, cos(z),
times the derivative of the first, 3z 2 . That is,
P
′ (z) = z
3 (− sin(z)) + cos(z)3z
2 = −z
3 sin(z) + 3z
2 cos(z).
The following activity further explores the use of the product rule.
Activity 2.7.
Use the product rule to answer each of the questions below. Throughout, be sure to
carefully label any derivative you find by name. It is not necessary to algebraically
simplify any of the derivatives you compute.
(a) Let m(w) = 3w 17 4 w . Find m ′ (w).
(b) Let h(t) = (sin(t) + cos(t))t 4 . Find h ′ (t).
(c) Determine the slope of the tangent line to the curve y = f (x) at the point
where a = 1 if f is given by the rule f (x) = e x sin(x).
(d) Find the tangent line approximation L(x) to the function y = g(x) at the point
where a = −1 if g is given by the rule g(x) = (x 2 + x)2 x .
⊳
change, while our (informal) discussion has also thought of this number as the total change in the number of
shares over the course of a single day. The formal proof of the product rule reconciles this issue by taking the
limit as the change in the input tends to zero.