What is Quotient Rule? Definition, Formula, Proof & Solved Examples

Quotient rule is a concept that is used in the chapter on calculus. It is a rule which is used to find the differentiation or the derivative of a function which is provided in the form of a ratio. We can also say that it is the rule to find out the differentiation between two differentiable functions. We can obtain the differentiation of a function which is in the form of u(x)/ v(x) where both u(x) and v(x) are differentiable functions and where v(x) is not equal to zero. If the value of v(x) turns out to be zero, the differentiation of the given function will become undefined. We take the help of the product rule while solving the sums using the quotient rule.

In this article, we will discuss the concepts of quotient rule and also take an example to understand it in a better way.

What is Quotient Rule? The Formula for the Quotient Rule in Differentiation

The formula for the quotient rule is given as follows:

f’(x) = [g(x) / h(x)]’ = [h(x) * g’(x) – g(x) * h’(x) / [h(x)] * [h(x)]

where,

f(x) is function for which the differentiation is to be obtained
f’(x) is the derivative of the function f(x)
g(x) is the function which is differentiable and forms part of the numerator of the function in consideration
h(x) is the function which is differentiable and forms part of the denominator of the function in consideration
g’(x) is the derivative of the function g(x)
h’(x) is the derivative of the function h(x)

Methods to Derive the Quotient Rule Formula

The quotient rule formula can be derived using a number of different methods. We can derive the quotient rule formula using:

Chain rule
Implicit differentiation
Derivative and properties of limits

Steps to Find Differentiation Using the Quotient Rule Formula

We can obtain the derivative of a function of the form g(x) / h(x) only if both the functions in consideration are differentiable. Follow the step given below to find the differentiation using the quotient rule formula:

Write down the values of g(x) and h(x) as given in the question.
Obtain the values of g’(x) and h’(x) and write them down.
Now apply the given quotient formula

f’(x) = [g(x) / h(x)]’ = [h(x) * g’(x) – g(x) * h’(x) / [h(x)] * [h(x)]

Let us take an example to understand the concept of quotient rule in a better way.

Find out the differentiation of the given function f(x) = 7x.x/ (x+1) using the quotient rule formula.

Solution: The given function for which differentiation needs to be found out is f(x) = 7x.x/ (x+1)

Here, g(x) = 7x.x and h(x) = x+1

Thus, g’(x) = 14x and h’(x) = 1

Now, we apply the formula given below:

f’(x) = [g(x) / h(x)]’ = [h(x) * g’(x) – g(x) * h’(x) / [h(x)] * [h(x)]

f’(x) = [ (x+1) * 14x – 7x.x * 1 / (x+1) (x+1)

f’(x) = 7xx + 14x / (x+1) (x+1)

Thus, f’(x) = 7xx + 14x / (x+1) (x+1) is the required answer.

Some basic tips that need to follow have been explained as follows:

Having a clear idea about the derivative and calculus rules in this industry is very much important so that people will be able to apply the right rule in the right situation without any kind of doubt. Categorising the functions and having a clear idea about the quotient rule implementation is very much important so the differentiation can be carried out very easily without any kind of doubt.
Spotting the products, quotient and compositions is another very important thing to be taken into consideration so that everybody will have a clear idea about the structuring of the function along with differentiating so that there is no chance of any kind of doubt in the whole process.
Taking the product of the derivatives is not same as applying the product rule which and taking the question of the derivatives is not same as applying the quotient rule. So, it is very much important for kids to be clear about the basic technicalities over here to avoid any kind of errors.
Normally students also confuse the function notation with the multiplication which is the main reason that being clear about all these kinds of technicalities is very much important because everything is entirely different function and differentiating it will result into the wrong derivative in the long run.
Re-writing the functions to make the differentiation very much easy is another very important aspect to be taken into consideration by the kids so that expressions are becoming much more efficient for the differentiation and further it will be the best possible opportunity of dealing with the things and making sure that there will be no chance of any kind of mistakes.
Re-writing the product as a simple polynomial is another very important thing to be taken into consideration so that application of the power rule can be carried out very easily and derivatives can be found without any kind of doubt. In this way it is very much vital for kids to be clear about the basic implementation of the question rule to avoid any kind of chaos in the whole process.

Being fluent in terms of taking the derivatives and having a clear idea about which rule has to be applied is very much important for kids in this particular area so that they can have a good hold over the entire chapter.

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