5128_Ch03_pp098-184

174 Chapter 3 Derivatives
EXPLORATION 1
Leaving Milk on the Counter
A glass of cold milk from the refrigerator is left on the counter on a warm summer
day. Its temperature y (in degrees Fahrenheit) after sitting on the counter t minutes is
y 72 300.98 t .
Answer the following questions by interpreting y and dydt.
1. What is the temperature of the refrigerator? How can you tell?
2. What is the temperature of the room? How can you tell?
3. When is the milk warming up the fastest? How can you tell?
4. Determine algebraically when the temperature of the milk reaches 55°F.
5. At what rate is the milk warming when its temperature is 55°F? Answer with
an appropriate unit of measure.
This equation answers what was once a
perplexing problem: Is there a function
with derivative x 1 ? All of the other
power functions follow the Power Rule,
d
x n nx n1 .
d x
However, this formula is not much help
if one is looking for a function with x 1
as its derivative! Now we know why: The
function we should be looking for is not
a power function at all; it is the natural
logarithm function.
Derivative of ln x
Now that we know the derivative of e x , it is relatively easy to find the derivative of its inverse
function, ln x.
y ln x
e y x Inverse function relationship
d
e d x
y d
x
d x
e y dy
1 d x
dy 1
d x e y 1 x
If u is a differentiable function of x and u 0,
Differentiate implicitly.
d
ln u 1 d x u d u
.
dx
2
1
0
y
(a, ln a)
slope = 1 a
–
1 2 3 4 5
y = ln x
Figure 3.57 The tangent line intersects
the curve at some point a, lna, where the
slope of the curve is 1a. (Example 3)
x
EXAMPLE 3 A Tangent through the Origin
A line with slope m passes through the origin and is tangent to the graph of y ln x.
What is the value of m?
SOLUTION
This problem is a little harder than it looks, since we do not know the point of tangency.
However, we do know two important facts about that point:
1. it has coordinates a, lna for some positive a, and
2. the tangent line there has slope m 1a (Figure 3.57).
Since the tangent line passes through the origin, its slope is
m ln a
a
0
0
ln a
a .
continued