In this explainer, we will learn how to find the derivative of a function using the quotient rule.
Once we have learned how to differentiate simple functions, we might start to wonder how we can differentiate more complex functions. Generally, more complex functions are created from simpler ones by combining them together in various ways. There are a few basic ways to combine two functions and :
- addition or subtraction: ;
- multiplication or division: or ;
- composition: .
To be able to differentiate more complex functions, it would be very helpful to have rules which tell us how to differentiate functions combined in these particular ways. At this point in a calculus course, we already know that the derivative of a sum is the sum of the derivatives:
Furthermore, we knowthat differentiation is actually a linear operation. This means that, in addition to the sum rule, we have the following rule for multiplication by a constant: where is a constant. Additionally, we have the product rule stated below.
Rule: Product Rule
Given two differentiable functions and , the derivative of their product is given by
This can be written succinctly using prime notation as follows:
In this explainer, we will focus on the rule for differentiating quotients. One way to think about a quotient is to think of it as a product of and . We will consider this approach first.
Example 1: Finding the Derivative of a Quotient Using the Product Rule
Find the first derivative of with respect to .
Answer
Let us consider as the product of two functions:
In doing this, we can apply the product rule, to find the derivative. Firstly, we need to find and using the power rule of differentiation: we have
Substituting the expressions for , , , and into the product rule, we have
By multiplying the numerator and denominator of the second fraction by , we can rewrite this as a single fraction:
Certainly, there is a simpler way to differentiate this function. We can begin by simplifying the expression for the function to
From this point, we can simply use the power rule to differentiate each term.
We can now generalize the method of expressing a quotient as a product to derive a general formula for the derivative of a quotient. We start by considering as the product of two functions and . We can now apply the product rule as follows:
Therefore, we only need to know how to evaluate and then we will have a general formula for derivatives of quotients. Let us consider the effect of a small change in on the value of . If changes by a small amount given by , there will be a corresponding change in which we denote . Therefore, the change in can be expressed as
Expressing this as a single fraction, we get
Dividing through by yields
Taking the limit as , we get the derivative of :
Using the properties of finite limits on continuous functions, we can rewrite this as
Since and , we have
We can now substitute this back into the product rule in equation (1), which gives us
Geometrically, we can visualize this as representing the area of a rectangle whose sides are of lengths and as shown in the figure.
By changing by a small value , the area of the rectangle will change. Assuming that both and are increasing, the change in will increase the area of the rectangle by , whereas the change in will decrease the area of the rectangle by . We will also need to subtract from the increase due to in order to find the area of the new rectangle.
Often, the quotient rule (equation (2)) is expressed as a single fraction as follows:
Unfortunately, this form can somewhat disguise the geometric interpretation. In spite of this, it is the form that is most often used in practice. Below is a statement of the quotient rule in full.
Rule: Quotient Rule
Given two differentiable functions and , the derivative of their quotient is given by
This can be written succinctly using prime notation as follows:
There is an alternative way to derive the quotient rule without appealing to the product rule. We can consider the changes in and are and , as a result of a small change in . Then, the corresponding change in is given by
Expressing this as a single fraction, we have
Dividing by , we have
Taking the limit as , we get the derivative of as follows:
Using the rules of finite limits for continuous functions, we have
Since , , and , we have
We will now consider a number of examples where we apply the quotient rule to find derivatives.
Example 2: Using the Quotient Rule to Find Derivatives
Find the first derivative of .
Answer
To find the derivative of , we will apply the quotient rule:
We set and . Before we can apply the quotient rule, we need to calculate the derivatives of and . Using the power rule of differentiation, we have
Substituting these expressions into the quotient rule, we have
Expanding the parentheses in the numerator, we have
Letโs consider another example for the quotient rule.
Example 3: Differentiating Quotient Functions
Differentiate .
Answer
We will apply the quotient rule, to find the derivative of . We begin by setting and . Now we need to find the derivatives of and . We can do this using the power rule as follows:
Substituting these expressions into the quotient rule, we have
We now expand the parentheses in the numerator and simplify as follows:
In the next example, we will find an unknown constant in a fractional expression when the value of the derivative of the expression at a point is given.
Example 4: Using the Quotient Rule
Suppose and . Determine .
Answer
Since we have been given the value of the derivative of at a particular point, we first need an expression for the derivative of . We can find such an expression using the quotient rule:
Setting and , we have
Substituting these expressions into the quotient rule, we have
To find the value of , we will use the fact that . Substituting into the expression for our derivative, we have
Hence,
Multiplying both sides of the equation by and dividing by yield
We can now expand the parentheses as follows:
Hence, by subtracting from both sides of the equation, we get
This expression can be factored by inspection, which gives alternatively, we could have used the quadratic formula or another method. Whatever approach taken, we get either solution of and .
In the next example, we will apply the quotient rule when the algebraic expressions for functions are not provided.
Example 5: Evaluating the Derivative at a Point Using the Quotient Rule
Let . Given that , , , and , find .
Answer
We begin by using the quotient rule, to find an expression for the derivative of . Let and . We begin by finding the derivatives of and as follows:
Substituting these into the quotient rule, we have
Setting , we have
Since , , , and , we have
Before applying the quotient rule, it is worth checking whether the given expression can be simplified. This is particularly important when the given expression involves a sum or a difference of quotients. Letโs consider an example where we simplify the given fractional expression before applying the quotient rule.
Example 6: Differentiating a Combination of Rational Functions Using the Quotient Rule
If , find .
Answer
When asked to differentiate a function like this, we could differentiate each term using the quotient rule. However, it is often simpler to express a sum of fractions as a single fraction and then apply the quotient rule once. This is the approach we will demonstrate here. We begin by rewriting the expression for as a single fraction:
We can now expand the parentheses in the numerator and denominator and simplify: We can now differentiate using the quotient rule, by setting and . We begin by finding the derivatives of and as follows:
We now substitute these expressions into the quotient rule to get
Letโs recap a few important concepts from this explainer.
Key Points
- To find the derivative of the quotient of two differentiable functions and , we can use the quotient rule which states that This is often written more succinctly using prime notation as follows:
- Before applying the quotient rule, it is worth checking whether it is possible to simplify the expression for the function.
