3.4: Differentiation Rules
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- State the constant, constant multiple, and power rules.
- Apply the sum and difference rules to combine derivatives.
- Use the product rule for finding the derivative of a product of functions.
- Use the quotient rule for finding the derivative of a quotient of functions.
- Extend the power rule to functions with negative exponents.
- Combine the differentiation rules to find the derivative of a polynomial or rational function.
Finding derivatives of functions by using the definition of the derivative can be a lengthy and, for certain functions, a rather challenging process. For example, previously we found that
by using a process that involved multiplying an expression by a conjugate prior to evaluating a limit.
The process that we could use to evaluate
In this section, we develop rules for finding derivatives that allow us to bypass this process. We begin with the basics.
The Basic Rules
The functions
The Constant Rule
We first apply the limit definition of the derivative to find the derivative of the constant function,
The rule for differentiating constant functions is called the constant rule. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is
Let
Alternatively, we may express this rule as
Find the derivative of
Solution
This is just a one-step application of the rule:
Find the derivative of
- Hint
-
Use the preceding example as a guide
- Answer
-
0
The Power Rule
We have shown that
At this point, you might see a pattern beginning to develop for derivatives of the form
Find
Solution:
|
Notice that the first term in the expansion of |
|
In this step the |
|
Factor out the common factor of |
|
After cancelling the common factor of |
|
Let |
Find
- Hint
-
Use
and follow the procedure outlined in the preceding example.
- Answer
-
As we shall see, the procedure for finding the derivative of the general form
Let
Alternatively, we may express this rule as
For
Since
we see that
Next, divide both sides by h:
Thus,
Finally,
□
Find the derivative of the function
Solution
Using the power rule with
Find the derivative of
- Hint
-
Use the power rule with
- Answer
-
The Sum, Difference, and Constant Multiple Rules
We find our next differentiation rules by looking at derivatives of sums, differences, and constant multiples of functions. Just as when we work with functions, there are rules that make it easier to find derivatives of functions that we add, subtract, or multiply by a constant. These rules are summarized in the following theorem.
Let
Sum Rule. The derivative of the sum of a function
that is,
Difference Rule. The derivative of the difference of a function
that is,
Constant Multiple Rule. The derivative of a constant
that is,
We provide only the proof of the sum rule here. The rest follow in a similar manner.
For differentiable functions
By substituting
Rearranging and regrouping the terms, we have
We now apply the sum law for limits and the definition of the derivative to obtain
□
Find the derivative of
Solution
We begin with the constant multiple rule, followed by the power rule:
Since

Find the derivative of
Solution
We begin by applying the rule for differentiating the sum of two functions, followed by the rules for differentiating constant multiples of functions and the rule for differentiating powers. To better understand the sequence in which the differentiation rules are applied, we use Leibniz notation throughout the solution:
Find the derivative of
- Hint
-
Use the preceding example as a guide.
- Answer
-
Find the equation of the line tangent to the graph of
Solution
To find the equation of the tangent line, we need a point and a slope. To find the point, compute
This gives us the point
so the slope of the tangent line is
Putting the equation of the line in slope-intercept form, we obtain
Find the equation of the line tangent to the graph of
- Hint
-
Use the preceding example as a guide.
- Answer
-
The Product Rule
Now that we have examined the basic rules, we can begin looking at some of the more advanced rules. The first one examines the derivative of the product of two functions. Although it might be tempting to assume that the derivative of the product is the product of the derivatives, similar to the sum and difference rules, the product rule does not follow this pattern. To see why we cannot use this pattern, consider the function
Let
That is,
This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
We begin by assuming that
By applying the limit definition of the derivative to
By adding and subtracting
After breaking apart this quotient and applying the sum law for limits, the derivative becomes
Rearranging, we obtain
By using the continuity of
□
For
Solution
Since
For
Solution
If we set
Simplifying, we have
To check, we see that
Use the product rule to obtain the derivative of
- Hint
-
Set
and and use the preceding example as a guide.
- Answer
-
The Quotient Rule
Having developed and practiced the product rule, we now consider differentiating quotients of functions. As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives; rather, it is the derivative of the function in the numerator times the function in the denominator minus the derivative of the function in the denominator times the function in the numerator, all divided by the square of the function in the denominator. In order to better grasp why we cannot simply take the quotient of the derivatives, keep in mind that
Let
That is, if
then
The proof of the quotient rule is very similar to the proof of the product rule, so it is omitted here. Instead, we apply this new rule for finding derivatives in the next example.
Use the quotient rule to find the derivative of
Solution
Let
Substituting into the quotient rule, we have
Simplifying, we obtain
Find the derivative of
- Hint
-
Apply the quotient rule with
and .
- Answer
-
It is now possible to use the quotient rule to extend the power rule to find derivatives of functions of the form
If
If
Simplifying, we see that
Finally, observe that since
□
Find
Solution
By applying the extended power rule with
Use the extended power rule and the constant multiple rule to find the derivative of
Solution
It may seem tempting to use the quotient rule to find this derivative, and it would certainly not be incorrect to do so. However, it is far easier to differentiate this function by first rewriting it as
Find the derivative of
- Hint
-
Rewrite
. Use the extended power rule with .
- Answer
-
.
Combining Differentiation Rules
As we have seen throughout the examples in this section, it seldom happens that we are called on to apply just one differentiation rule to find the derivative of a given function. At this point, by combining the differentiation rules, we may find the derivatives of any polynomial or rational function. Later on we will encounter more complex combinations of differentiation rules. A good rule of thumb to use when applying several rules is to apply the rules in reverse of the order in which we would evaluate the function.
For
Solution: Finding this derivative requires the sum rule, the constant multiple rule, and the product rule.
Apply the sum rule. | |
Apply the constant multiple rule to differentiate |
|
For
Solution
We can think of the function
For
Solution
This procedure is typical for finding the derivative of a rational function.
Find
- Hint
-
Apply the difference rule and the constant multiple rule.
- Answer
-
Determine the values of
Solution
To find the values of
Since
we must solve

The position of an object on a coordinate axis at time
Solution
Since the initial velocity is
After evaluating, we see that
Find the values of
- Hint
-
Solve
.
- Answer
-
Formula One car races can be very exciting to watch and attract a lot of spectators. Formula One track designers have to ensure sufficient grandstand space is available around the track to accommodate these viewers. However, car racing can be dangerous, and safety considerations are paramount. The grandstands must be placed where spectators will not be in danger should a driver lose control of a car (Figure

Safety is especially a concern on turns. If a driver does not slow down enough before entering the turn, the car may slide off the racetrack. Normally, this just results in a wider turn, which slows the driver down. But if the driver loses control completely, the car may fly off the track entirely, on a path tangent to the curve of the racetrack.
Suppose you are designing a new Formula One track. One section of the track can be modeled by the function

- Physicists have determined that drivers are most likely to lose control of their cars as they are coming into a turn, at the point where the slope of the tangent line is 1. Find the
coordinates of this point near the turn. - Find the equation of the tangent line to the curve at this point.
- To determine whether the spectators are in danger in this scenario, find the
-coordinate of the point where the tangent line crosses the line . Is this point safely to the right of the grandstand? Or are the spectators in danger? - What if a driver loses control earlier than the physicists project? Suppose a driver loses control at the point (
). What is the slope of the tangent line at this point? - If a driver loses control as described in part 4, are the spectators safe?
- Should you proceed with the current design for the grandstand, or should the grandstands be moved?
Key Concepts
- The derivative of a constant function is zero.
- The derivative of a power function is a function in which the power on
becomes the coefficient of the term and the power on in the derivative decreases by 1. - The derivative of a constant
multiplied by a function is the same as the constant multiplied by the derivative. - The derivative of the sum of a function
and a function is the same as the sum of the derivative of and the derivative of . - The derivative of the difference of a function
and a function is the same as the difference of the derivative of and the derivative of . - The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.
- The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.
- We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.
Glossary
- constant multiple rule
- the derivative of a constant
multiplied by a function is the same as the constant multiplied by the derivative:
- constant rule
- the derivative of a constant function is zero:
, where is a constant
- difference rule
- the derivative of the difference of a function
and a function is the same as the difference of the derivative of and the derivative of :
- power rule
- the derivative of a power function is a function in which the power on
becomes the coefficient of the term and the power on in the derivative decreases by 1: If is an integer, then
- product rule
- the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function:
- quotient rule
- the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function:
- sum rule
- the derivative of the sum of a function
and a function is the same as the sum of the derivative of and the derivative of :