Short-Cuts to Differentiation

3.6	The Chain Rule and Inverse Functions

In this section we will use the chain rule to calculate the derivatives of fractional powers, logarithms, exponentials, and the inverse trigonometric functions.⁵ The same method is used to obtain a formula for the derivative of a general inverse function.

Finding the Derivative of an Inverse Function: Derivative of

Earlier we calculated the derivative of

with

an integer, but we have been using the result for non-integral values of

as well. We now confirm that the power rule holds for

by calculating the derivative of

using the chain rule. Since

the derivative of

and the derivative of

must be equal, so

We can use the chain rule with

as the inside function to obtain:

Solving for

gives

A similar calculation can be used to obtain the derivative of

where

is a positive integer.

Derivative of

We use the chain rule to differentiate an identity involving

. Since

, we have

Solving for

gives

Example 1

Differentiate

(a)

(b)

(c)

Solution

(a)

Using the chain rule:

(b)

Using the product rule:

(c)

Using the chain rule:

Derivative of

In Section 3.2, we saw that the derivative of

is proportional to

. Now we see another way of calculating the constant of proportionality. We use the identity

Differentiating both sides, using

and the chain rule, and remembering that ln

is a constant, we obtain:

Solving gives the result we obtained earlier:

Derivatives of Inverse Trigonometric Functions

In Section 1.5 we defined arcsin

as the angle between

and

(inclusive) whose sine is

. Similarly,

as the angle strictly between

and

whose tangent is

. To find

we use the identity

. Differentiating using the chain rule gives

Using the identity

, and replacing θ by

, we have

Thus we have

By a similar argument, we obtain the result:

Example 2

Differentiate

(a)

(b)

Solution

(a)

Use the chain rule:

(b)

Use the chain rule:

Derivative of a General Inverse Function

Each of the previous results gives the derivative of an inverse function. In general, if a function

has a differentiable inverse,

, we find its derivative by differentiating

by the chain rule:

Thus, the derivative of the inverse is the reciprocal of the derivative of the original function, but evaluated at the point

instead of the point

Example 3

Figure 3.29 shows

and

. Using Table 3.6, find

(a)

Figure 3.29

Table 3.6


0	1	0.7
1	2	1.4
2	4	2.8
3	8	5.5


(i)

(ii)

(iii)

(iv)

(b)	The equation of the tangent lines at the points and .

(c)	What is the relationship between the two tangent lines?

Solution

(a)

Reading from the table, we have

(i)

(ii)

(iii)

(iv)

To find the derivative of the inverse function, we use

Notice that the derivative of

is the reciprocal of the derivative of

. However, the derivative of

is evaluated at 2, while the derivative of

is evaluated at 1, where

and

(b)

At the point

, we have

and

, so the equation of the tangent line at

At the point

, we have

, so the slope at

Thus, the equation of the tangent line at

(c)	The two tangent lines have reciprocal slopes, and the points and are reflections of one another in the line . Thus, the two tangent lines are reflections of one another in the line .

Exercises and Problems for Section 3.6

Exercises

For Exercises 1-41, find the derivative. It may be to your advantage to simplify before differentiating. Assume

, and

are constants.

9.	2 Since , the derivative .

10.

11.

12.

13.

14.

15.

16.

17.	Using the product and chain rules gives .

18.

19.

20.

21.	(because or because ), so .

22.

23.

24.

25.

26.

27.

28.

29.	Note that so, .

30.

31.

32.

33.

34.

35.

36.

37.	Using the chain rule gives .

38.

39.

40.

41.

Problems

42.

Let

(a)	Find and simplify your answer.

(b)	Use properties of logs to rewrite as a sum of logs.

(c)	Differentiate the result of part (b). Compare with the result in part (a).

43.	On what intervals is concave up?

44.	Use the chain rule to obtain the formula for .

45.

Using the chain rule, find

. (Recall

Let

Then

Differentiating,

46.

To compare the acidity of different solutions, chemists use the pH (which is a single number, not the product of

and

). The pH is defined in terms of the concentration,

, of hydrogen ions in the solution as

Find the rate of change of pH with respect to hydrogen ion concentration when the pH is 2.

47.

The number of years,

, it takes an investment of $1000 to grow to

in an account which pays 5% interest compounded continuously is given by

Find

and

. Give units with your answers and interpret them in terms of money in the account.

48.

A firm estimates that the total revenue,

, in dollars, received from the sale of

goods is given by

The marginal revenue,

, is the rate of change of the total revenue as a function of quantity. Calculate the marginal revenue when

49.

(a)	Find the equation of the tangent line to at . For , we have , so the slope of the tangent line is . The equation of the tangent line is , so, on the tangent line, .

(b)

Use it to calculate approximate values for

and

Using a value on the tangent line to approximate

, we have

Similarly,

is approximated by

(c)

Using a graph, explain whether the approximate values are smaller or larger than the true values. Would the same result have held if you had used the tangent line to estimate

and

? Why?

From Figure 3.3, we see that

and

are below

. Similarly,

and

are also below

. This is true for any approximation of this function by a tangent line since

is concave down

. Thus, for a given

-value, the

-value given by the function is always below the value given by the tangent line.

Figure 3.3

50.

Find the equation of the best quadratic approximation to

. The best quadratic approximation has the same first and second derivatives as

(a)	Use a computer or calculator to graph the approximation and on the same set of axes. What do you notice?

(b)	Use your quadratic approximation to calculate approximate values for and .

51.

(a)	For , find and simplify the derivative of .

(b)	What does your result tell you about ? is a constant function

52.

Imagine you are zooming in on the graph of each of the following functions near the origin:

Which of them look the same? Group together those functions which become indistinguishable, and give the equations of the lines they look like.

In Problems 53-56, use Figure 3.30 to find a point

where

has the given derivative.

Figure 3.30

53.

Since the chain rule gives

we must find values

and

such that

and

Calculating slopes from the graph of

gives

Calculating slopes from the graph of

gives

The only values of the derivative

are 1 and

and the only values of the derivative

are 2 and

. In order to have

we must therefore have

and

. Thus

and

Now

and from the graph of

we see that

for

The two conditions on

we have found are both satisfied when

. Thus

for all

in the interval

. The question asks for just one of these

values, for example

54.

55.

56.

In Problems 57-59, use Figure 3.31 to estimate the derivatives.

Figure 3.31

57.

58.

59.

In Problems 60-62, use Figure 3.32 to calculate the derivative.

Figure 3.32

60.

61.

62.

63.

(a)	Given that , find . 12

(b)

Find

(c)	Use your answer from part (b) to find .

(d)	How could you have used your answer from part (a) to find ?

64.

(a)	For , find .

(b)	How can you use your answer to part (a) to determine if is invertible?

(c)

Find

(d)

Find

(e)

Find

65.

Use the table and the fact that

is invertible and differentiable everywhere to find


3	1	7
6	2	10
9	3	5

66.

At a particular location,

is the number of gallons of gas sold when the price is

dollars per gallon.

(a)	What does the statement tell you about gas sales?

(b)	Find and interpret .

(c)	What does the statement tell you about gas sales?

(d)	Find and interpret

67.

Let

give the US population⁶ in millions in year

(a)	What does the statement tell you about the US population? Pop is 296 m in 2005

(b)	Find and interpret . Give units. 2005

(c)	What does the statement tell you about the population? Give units. Pop incr by

(d)	Evaluate and interpret . Give units.

68.

Figure 3.33 shows the number of motor vehicles,⁷

, in millions, registered in the world

years after 1965. With units, estimate and interpret

(a)

(b)

(c)

(d)

Figure 3.33

69.

Using Figure 3.34,

, find

Figure 3.34

70.

is increasing and

, which of the two options, (a) or (b), must be wrong?

(a)

(b)

71.

An invertible function

has values in the table. Evaluate

(a)

(b)

(c)

72.	If is continuous, invertible, and defined for all , why must at least one of the statements , be wrong?

73.

(a)

Calculate

by identifying the limit as the derivative of

(b)

Use the result of part (a) to show that

The rules of logarithms give

Thus, taking

to both sides and using the fact that

, we have

This limit is sometimes used as the definition of

(c)

Use the result of part (b) to calculate the related limit,

Let

. Then as

, we have

. Since

we have

This limit is also sometimes used as the definition of

Strengthen Your Understanding

In Problems 74-76, explain what is wrong with the statement.

74.

then

75.

The derivative of

76.

Given

, and

, we have

In Problems 77-80, give an example of:

77.	A function that is equal to a constant multiple of its derivative but that is not equal to its derivative. Any exponential function of the form with , and is a constant multiple of its derivative but it is not equal to its derivative: . One example is .

78.	A function whose derivative is , where c is a constant.

79.	A function for which , where is a constant.

80.	A function such that .

Are the statements in Problems 81-82 true or false? Give an explanation for your answer.

81.

The graph of

is concave up for

82.	If has an inverse function, , then the derivative of is .