The Quotient Rule says that the derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator then all divided by the square of the denominator.

Reason for the Quotient Rule:

The Product Rule should be utilized when the derivative of the quotient of two functions is to be taken.

Method of Finding the Derivatives of Two Functions  

It is expressed as:

  • The denominator function times 
  • The derivative of the numerator function
  • The numerator function times
  • The derivative of the denominator function
  • The square of the denominator function

Quotient Rules Calculus Examples

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

This rule always starts with a denominator function and ends up with a denominator function.

Easy Steps to Find the Derivatives using the Quotient Rule

  • Considering the given function, it should be in the form of division.
  • Differentiating both sides of the function with respect to something.
  • Suppose if the function on LHS is y equal to some other function of x.
  • The derivatives on the RHS will be the lower function times the derivative of the upper function minus the upper function times the derivative of the lower function divided by the square of the lower function.
  • The answer you will get at the end after the simplification will be the derivative of the function y which will be given to you.

Application of Derivatives in Real Life

Have you ever imagined that you will be using quotient rules in real life? If not, then let me tell you how. 

Some examples are:-

  • Calculating profit and loss in business using graphs.
  • To check the temperature variations.
  • Determining the speed or distance covered such as miles per hour, kilometer per hour, etc.
  • Derivatives are also used to derive many equations in Physics.
  • In the study of Seismology, to find the range of magnitudes of the earthquake.

Derivative Rules

There are three very important rules that are generally applicable and they depend on the structure of the function we are differentiating:

  • The Product
  • Quotient
  • Chain Rules

Quotient Rule – Example

The quotient is the answer obtained when we divide one number with another. 

For example, when dividing the number 6 by 3, we get the result as 2, which is the quotient. The quotient can also be an integer or a decimal number.

There are so-called “shortcut” rules for finding the derivative of a function. The quotient rule is a rule used to find the derivative of a function that can be written as the quotient of two functions. More simply, you can think of the quotient rule before applying to functions that are written out as fractions where the numerator and the denominator are both themselves functions.

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