 # Quotient Rule and the Differences Between Quotient Rule and Product Rule

February 7, 2022

We learn about the concept of quotient rule in the chapter on differentiation and integration. Most of the students find this concept very difficult. However, once you understand its concept clearly, things become very easy for you. Most of us know that we come across the word ‘quotient’ in the arithmetic operation of division. In the chapter on differentiation, we use the quotient rule to find out the derivative of a given function which is in the form of a ratio. In a layman’s language, we can say that the quotient rule is a rule which is used to find out the derivative of two functions that are differentiable. It is important in the quotient rule that the denominator of the function that needs to be differentiated should not be equal to zero. In this article, we will try to understand the concepts of quotient rule clearly along with discussing the differences between the quotient rule and the product rule.

## Quotient Rule and its Formula

We can obtain the value of the derivative of two functions which are in the form of a ratio using the quotient rule. We have the formula for the quotient rule which is given below:

f'(x) = [g(x)/k(x)]’ = [k(x) × g'(x) – g(x) × k'(x)]/[k(x)]2

Where,

f(x) is the function for which we need to obtain the derivative using the quotient rule.

f'(x) is the derivative of the function that is obtained using the quotient rule.

g(x) is the numerator part of f(x) which is also differentiable.

g'(x) is the derivative of the function g(x)

k(x) is the denominator part of f(x) which is also differentiable.

and k'(x) is the derivative of the function k(x).

## What are Different Processes by Which We Can Derive Quotient Rule?

We can obtain or derive the formula for the quotient rule using various processes which are discussed below:

a) Using the process of the chain rule

b) Using the process of implicit differentiation

c) Using the process derivative and limit properties.

## Differences Between Quotient Rule and Product Rule

We use the product rule and the quotient rule to find out the derivative of various functions. When there is the presence of multiplication of two or more numbers, then we take the help of the product rule to find out the derivative. Whereas, where there is a division of two differentiable functions, then we take the help of the quotient rule to find out the derivative. We can say that the quotient rule and the product rule are very similar in a way. The formula of the product rule can be seen in the formula of the quotient rule.

Let us understand where to use the product rule and the quotient rule by taking some examples. Suppose we have a function as f(x) = 37 x. x cos x. We can see that this function comprises the multiplication of two different functions. Thus, we will use the product rule here. Now, suppose we have a function as f(x) = 37x.x / cos x. We can see that this function comprises the division of two different functions. Thus, we will use the quotient rule here.

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