What is the Quotient Rule?


What is the Quotient Rule?

Quotient rule is an important topic in Mathematics. Let’s learn about it in this article. Quotient Rule is a method of finding out the derivative of a function which is given in the form of the ratio of two differentiable functions. The quotient rule is a rule in calculus for taking the derivative of a quotient of two functions. It is the most important topic of differentiation.

Let g(x) and h(x) are two differentiable functions and h(x) is not equal to 0.

Then, we can apply the Quotient rule in the form of f(x). i.e; 

f(x)= g(x)/h(x), d(f(x)/g(x))= (g(x) x ff(x) – f(x) x dg(x)) / (g(x))2

Here, d denotes a derivative.

          df(x) is derivative of function f.

          dg(x) is derivative of function g.

For finding the derivative of f(x) divided by g(x), you must:

Take g(x) times the derivative of f(x). Then from that product, you must subtract the product of f(x) and the derivative of g(x). And, finally divide those terms by g(x) squared. For more detailed knowledge you can join Cuemath classes.

Steps to Find The Derivatives Using The Quotient Rule

  1. First check the given function, it should be in the form of division.
  2. Differentiate both sides of the function.
  3. Let the function on the Left Hand Side be y and equal to some other function of x.  Then, derivatives of the Left Hand Side will be dy/dx.
  4. The derivatives on the Right Hand Side will be denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. At the end, the answer you will get after simplification will be the derivative of the function y which is given to you. To learn more about the quotient rule, join classes.

Quotient Rule Examples

Quotient rule is quite similar to the product rule in calculus. The only difference between the product rule and the quotient rule is that in the product rule we have the function in the product form of type f(x)*g(x) and in the quotient rule, we have the function in the division form of type f(x)/g(x).

Examples are 3x/x^4, e^3x/2x, 4x^2/x^5, etc.

Example :

Consider a function f(x) which is given by,

f(x) = g(x) /  h(x)

Then the derivative of the function f(x) is as follows:-

f'(x)= g(x)/h(x)’

The quotient rule gives us derivatives of a function in a unique way, It is the combination of the original function and derivatives of that given function. The integral quotient rule is the method of integrating two functions given in the form of numerator and denominator(p/q). This rule is also called the division rule or Antiderivative quotient.


If a function is a product, sum, or quotient of a simpler function, then we can use the product, sum, or quotient rule to differentiate the overall function in terms of simpler functions and their derivatives.

  • The product rule states that if P is a product of two functions g and h which are differentiable according to the rule  P(x)=g(x)h(x), then


  • If  Q  is a quotient of differentiable functions  g  and  h  then according to the quotient rule Q(x) = g(x) h(x) , then



Remember that the quotient rule begins with the bottom function or the denominator and ends with the bottom function squared.


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