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The Quotient Rule is one of the most helpful tools in Differential Calculus (or Calculus I) to derive two functions that are being divided. It can be used along with any existing types of functions as long as division operations are present within the given derivation problem.

Here, we will focus mainly on the proofs of the quotient rule formula by applying the concepts of derivation through limits and the chain rule. Also, we will look at some derivation examples of functions that use the quotient rule formula.

CALCULUS

Relevant for

Learning about the different proofs of the quotient rule.

See proofs

CALCULUS

Relevant for

Learning about the different proofs of the quotient rule.

See proofs

What is the the Quotient Rule?

The quotient rule is defined as the derivative of the quotient of functions. The quotient rule can be used to derive any given quotient of functions such as but not limited to:

where f(x) and g(x) can be equivalent to any types of functions.

But how exactly do we derive that given function using the quotient rule?

The quotient rule states that the derivative of a quotient of functions is equal to the quantity of the denominator g(x) multiplied by the derivative of the numerator f(x) subtracted to the numerator f(x) multiplied by the derivative of the denominator g(x), all divided by the square of the denominator g(x).

To better illustrate, when you are given two functions f(x) and g(x) and then you are asked to get the derivative of or the derivative of the quotient of f(x) and g(x), we have:

Seems easy, right? But we should not take this formula superficially if we aim to be able to derive any quotient of functions. In order to learn and understand the concepts behind the development of this quotient rule formula, we need to be familiarized with any proof which would satisfy the statement of the quotient rule.

Proof of The Quotient Rule Using Limits

In this article, you are highly recommended to be familiarized with the topics The Slope of a Tangent Line and Derivatives Using Limits, as a pre-requisite to better understand the proof of the quotient rule using limits.

We can recall that

Now, we are going to use the following expression:

Then we have,

Using limits, we can derive by

By substituting the equation , we have

By getting the least common denominator of the numerator, we have

By applying the rules for fractions, our equation can be re-written as:

Now, we can add and subtract the product of f(x) and g(x), which is , to the numerator . Hence, we have

Given that , we didn’t change the equation at all.

Re-arranging the previous equation, we have

Now, we can further simplify the previous equation by factoring the numerator:

Then we can further re-arrange the equation like this:

so that we can algebraically manipulate it in a way necessary to prove the quotient rule.

By applying the properties of limits to solve the equation, we have

Then, we can solve the limits by recognizing that the first part of each term is simply equal to the functions and respectively and the second part of each term is the limit derivative of and respectively. Therefore, we have:

By simplifying algebraically, we have

or it can be simply illustrated as

which is now The Quotient Rule Formula.

Proof of The Quotient Rule Using The Product andChain Rules

Another way that might make the quotient rule easier to prove and formulate is by applying the product and chain rules’ formulas. Hence, you are highly recommended to be familiarized with the topics, The Chain Rule Formula and The Product Formula as a pre-requisite to better understand this proof.

We can recall that the product rule formula is

In addition to the product rule, we also recall that the chain rule formula is

Now, if we are given two functions f(x) and g(x) and then, we are asked to get the derivative of ; we have

By re-writing the denominator of the fraction into exponential form, we have

Now, we can derive the right hand side of the equation by applying the product rule formula:

To derive , we need to use the chain rule formula. Hence we have

By applying all applicable operations, getting the least common denominator, re-writing the negative exponent into fractional form, and following the rules of fractions, we have

Then finally, we get

which is now The Quotient Rule Formula.

Proof of The Quotient Rule Using Implicit Differentiation

This is actually the shortest method of proving the quotient rule formula considering you are familiarized with the topics, the Product Rule, and Implicit Differentiation.

We can recall that implicit differentiation is used for functions in a more complicated form in which it is difficult or impossible to express f(x) or y explicitly in terms of x. For instance, we are given an equation:

and then we are asked to derive . By deriving , we have

But assuming we cannot further simplify our equation algebraically and still do not know the quotient rule formula, we can do derive it by applying implicit differentiation.

To implicitly differentiate our given equation, we must first algebraically cross multiply the denominator of the right-hand side to the left-hand side of the equation. Hence, we have

Then, we derive the whole equation in terms of the variable :

To derive the left-hand side of the equation, we will use the product rule. We will also treat and as variables and not constants. By doing so, we have

How about ? How do we derive in terms of variable in this case? Just like and , we will treat as a variable, not as a constant. By doing so, we have

Since we are asked to get , we need to equate our equation in terms of . By doing so, we have

But what is ? We can recall from the beginning that . By substituting to our derived equation, we have

By simplifying algebraically, getting the least common denominator, and applying the rules of fractions, we have

Then finally, we have

which is now The Quotient Rule Formula.

Quotient Rule – Examples with answers

The following examples can be used to learn how to apply the quotient rule formula to find derivatives.

EXAMPLE 1

Derive the following:

The first thing we need to do is to list down the quotient rule formula for our reference:

We have as the numerator/dividend and as the denominator/divisor.

Based on the quotient rule formula, let be the numerator and be the denominator. Therefore, we have

After doing this, we can now use the quotient rule formula to derive our given problem:

Next, let’s derive and individually and then substitute it to the quotient rule formula later:

∴  used the power formula

∴  used the power formula, sum/difference of derivatives, and derivatives for constants

By substituting , , , and into the quotient rule formula, we have:

Simplifying algebraically, we get

EXAMPLE 2

What is the derivative of the following?

Let’s start by listing down the quotient rule formula for our reference:

Based on our given, we have as the numerator/dividend and as the denominator/divisor. Therefore, we have

After doing this, we can now use the quotient rule formula to derive our given problem:

Next, let’s derive and individually and then substitute it to the quotient rule formula later:

∴  used the power formula

∴  used the derivative for logarithmic functions

By substituting , , , and into the quotient rule formula, we have:

Simplifying algebraically, we get

EXAMPLE 3

Derive the following:

Let be the numerator and be the denominator. Therefore, we have

After doing this, we can now use the quotient rule formula to derive our given problem:

Next, let’s derive and individually and then substitute it to the quotient rule formula later:

∴  used the power formula and the sum/difference of derivatives

∴  used the power formula and the sum/difference of derivatives

By substituting , , , and into the quotient rule formula, we have:

Simplifying algebraically, we get

EXAMPLE 4

What is the derivative of f(x)?

Let be the numerator and be the denominator. Therefore, we have

After doing this, we can now use the quotient rule formula to derive our given problem:

Next, let’s derive and individually and then substitute it to the quotient rule formula later:

∴  used the derivative for trigonometric functions

∴  used the derivative for trigonometric functions

By substituting , , , and into the quotient formula, we have:

Simplifying algebraically and by applying trigonometric identities, we get

EXAMPLE 5

Derive f(x):

Based on our given, we have as the numerator/dividend and as the denominator/divisor.

Based on the quotient rule formula, we have as the numerator and as the denominator. Therefore, we have

When deriving roots, it is always advisable to re-write radicals into exponential form. Hence,

After doing this, we can now use the quotient rule formula to derive our given problem:

Next, let’s derive and individually and then substitute it to the quotient rule formula later:

∴  used the power formula

∴  used the power formula and the sum/difference of derivatives

By substituting , , , and into the quotient rule formula, we have:

Simplifying algebraically, we get