Quotient Rule Calculator

If you want to evaluate the quotient rule having the derivative of two ratio functions? Then try our quotient rule calculator to determine the rational function.

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Table of Contents:

Introduction To Quotient Rule Calculator:

Quotient rule calculator is an online tool that helps you to evaluate the derivative of two ratio functions in a few seconds. It is used to determine the rational function with the help of the quotient rule of differentiation of n order.

Quotient Rule Calculator with Steps

Quotient rule derivative calculator is an amazing tool that keeps you away from complex manual calculation of quotient derivation problems and gives you solutions of simple or complicated both types of quotient rule questions quickly.

What Is Quotient Rule?

The quotient rule is a rule that is used to solve differentiation of rational function in calculus. When you have two ratio function f(x) and g(x) then the differentiation of function h(x) is given as,

$$ h(x) \;=\; \frac{f(x)}{g(x)} $$

The quotient rule for differentiation of function h(x) is,

$$ h’(x) \;=\; \frac{f’(x) g(x) - f(x) g’(x)}{[g(x)]^2} $$

Whereas f(x) is the numerator of a given function and its differentiation is f`(x) and the denominator function is g(x) and differentiation is g`(x).

How To Find Derivative Quotient Rule?

To find the derivative of a ratio of two functions f(x) and g(x) we use the quotient rule of differentiation.

You can easily solve the simple quotient rule problem but some problems are complex and cannot be easily solved by hand especially when you are not familiar with the derivation concept. Here is a complete guide about how to apply the quotient rule step-by-step that helps you to solve any type of quotient rule problem without any difficulty.

Step 1:

Identify the given function h(x) defined as the ratio of two functions f(x) and g(x):

$$ h(x) \;=\; \frac{f(x)}{g(x)} $$

Step 2:

To find the derivative h′(x) the quotient rule is used such as:

$$ h’(x) \;=\; \frac{f’(x) g(x) - f(x) g’(x)}{[g(x)]^2} $$

Step 3:

Identify the numerator function f(x) and denominator function g(x).

Step 4:

Differentiate the numerator function f(x) and the denominator function g(x).

Step 5:

Add the value of f`(x) and g`(x) in the quotient rule formula for further calculation.

Step 6:

Simplify the above expression after adding the differential value of both the function and you get the solution of the quotient rule problem.

Solved Example Of Quotient Rule:

An example is given below to let you understand how the quotient rule solves the quotient rule problems easily.

Example: find the derivative of:

$$ h(x) \;=\; \frac{x^2 + 3}{x - 1} $$

Solution:

Step 1: Identify the numerator and denominator functions.

Step 2: Differentiate the numerator function f(x).

Step 3: Differentiate the denominator function g(x).

Step 4: Add the quotient rule value and put in the quotient rule formula.

Step 5: Simplify the above expression to get the solution of quotient rule.

By following these steps, you can apply the quotient rule to find the derivative of any function that is in the ratio of two functions.

How To Use Quotient Rule Derivative Calculator?

The derivative calculator quotient rule has a simple design that helps you to solve the given rational differential function immediately. You just need to put your given ratio function in this derivative calculator using quotient rule with steps only by following some simple steps. These steps are:

  • Enter the ratio function to find its differentiation in the input box.
  • Choose the variable of differentiation for the given ratio function in the input field.
  • Check your given quotient function to get the exact solution of the quotient rule question.
  • Click on the Calculate button to get the result of the given ratio derivative problems.
  • If you want to check the working procedure of the derivative quotient rule calculator then you can use the given example to get a solution.
  • The “Recalculate” button for the calculation of more examples of quotient rule function with the solution.

Final Result Of Derivative Derivative Calculator Quotient Rule:

Quotient derivative calculator provides you with a solution as per your input problem when you click on the calculate button. It may include as:

In the Result Box:

Click on the result button so you get the solution of your quotient rule derivative calculator question.

Steps Box:
When you click on the steps option, you get the result of the ratio function of differentiation in a step-by-step process.

Advantages of Derivative Using Quotient Rule Calculator:

The derivative calculator quotient rule has many advantages when you use it to find the solution of a given ratio function. Our tool only gets the input value and it provides a solution without taking any external assistance. These advantages are:

  • It is a trustworthy tool as it always provides you with accurate solutions to given quotient rule derivative problems.
  • Quotient derivative calculator is an efficient tool that provides solutions to the given indefinite integral problems in a few seconds.
  • It is a learning tool that provides you in depth knowledge about the concept of quotient derivative function very easily on an online platform.
  • It is a handy tool that evaluate various types of ratio function for differential problems quickly without manual calculation.
  • Derivative calculator using quotient rule is a free tool that allows you to use it for the calculation of quotient rule derivative questions without paying.
  • Quotient rule derivative calculator is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of differential quotient problems.
Related References
Frequently Ask Questions

What is the difference between product rule and quotient rule

The product rule and the quotient rule are both differentiation rules used in calculus to find the derivatives of different types of functions. The differences between them are:

Product Rule

The product rule is a derivative used to calculate the product of two functions. If you have two functions u(x) and v(x), and you want to find the derivative of their product w(x)=u(x)⋅v(x), the product rule states that,

$$ w′(x) \;=\; u′(x) v(x) + u(x) v′(x) $$

Quotient Rule

The quotient rule is the derivative rule that is used to calculate the ratio of two functions. If you have a function h(x) defined as the ratio of two functions f(x) and g(x):

$$ h(x) \;=\; \frac{f(x)}{g(x)} $$

The quotient rule states:

$$ h′(x) \;=\; \frac{f′(x)g(x) − f(x)g′(x)}{g(x)^2} $$

Hence they both have different methods to solve derivatives of different functions.

How to Derive the Formula for Quotient Rule?

The quotient rule formula derivation is:

Consider a function

$$ h(x) \;=\; \frac{f(x)}{g(x)} $$

The definition of derivation is,

$$ f’(a) \;=\; \lim_{x \to 0} \frac{f(a+h) - f(a)}{h} $$

Add the above function value to it,

$$ h’(x) \;=\; \lim_{h \to 0} \frac{f(x + h) g(x) - f(x) g(x + h)}{\frac{g(x + h) g(x)}{h}} $$

$$ h’(x) \;=\; \lim_{h \to 0} \frac{f(x + h) g(x) - f(x) g(x + h)}{h . g(x + h) g(x)} $$

Separate the function after simplification,

$$ h’(x) \;=\; \lim_{h \to 0} \frac{g(x) [x + h) - f(x)] + f(x) [g(x) - g(x + h)]}{h . g(x + h) g(x)} $$

$$ h’(x) \;=\; \lim_{h \to 0} \left( \frac{g(x) [f(x + h) - f(x)]}{h . g(x + h) g(x)} \right) + \lim_{h \to 0} \left( \frac{f(x) [g(x) - g(x +h)}{h . g(x + h) g(x)} \right) $$

For first function,

$$ \lim_{h \to 0} \left( \frac{g(x) [f(x + h) - f(x)]}{h . g(x+h) g(x)} \right) \;=\; \frac{g(x) f’(x)}{g(x) g(x)} \;=\; \frac{f’(x) g(x)}{g(x)^2} $$

For second function,

$$ \lim_{h \to 0} \left( \frac{f(x) [g(x) - g(x + h)]}{h . g(x + h) g(x)} \right) \;=\; \lim_{h \to 0} \left( \frac{f(x) [- (g(x + h) - g(x))]}{h . g(x + h) g(x)} \right) \;=\; \frac{f(x) g’(x)}{g(x) g(x)} $$

Combine both the parts,

$$ h’(x) \;=\; \frac{f’(x) g(x)}{g(x)^2} - \frac{f(x) g’(x)}{g(x)^2} $$

After solving and rearranging the above expression we get the quotient rule such as,

$$ h’(x) \;=\; \frac{f’(x) g(x) - f(x) g’(x)}{g(x)^2} $$

What are some common mistakes when using the quotient rule?

Some ratio functions are very complex when you apply the quotient rule to it, the derivation process becomes more complicated. In this situation, you may commit some mistakes in quotient rule derivation and you cannot find the accurate solution.

  • Incorrect use of the Rule:

Some differential problems are difficult to recognize whether to use the product rule or quotient rule so you may select the incorrect method for the solution.

  • Mixing Up f(x)) and g(x):

When you treat with a similar function like sinx/cosx function, the derivation of sinx is cosx and cosx is sinx. You can mix these functions because of confusion.

  • Algebraic Errors:

You may commit a mistake in algebraic expression after differentiation

  • Ignoring the Denominator Squared:

In the quotient rule problem, the denominator has taken a square value but you may ignore the square because the given function g(x) has a square value of x^2 instead of x^4.

  • Not Simplifying the Final Answer:

You may not simplify the final result after the differentiation of a given function.

To avoid these common mistakes and carefully follow the steps of the quotient rule, you can avoid errors and correctly find the derivative of a quotient of functions.

What is the quotient rule used for

The quotient rule is used in various types of fields which is given as

  • Differentiating Rational Functions:

In mathematical calculus, the quotient rule is used to differentiate rational functions.

  • Solving Problems in Physics and Engineering:

In physics and engineering, for ratio quantities, the quotient rule is used.

  • Economics and Business:

In economics and business, functions representing cost, revenue, and profit has also ratios function so the quotient rule is used.

  • Biological and Medical Sciences:

The quotient rule can be used to model and analyze rates and ratios, such as rates of enzyme reactions or population growth rates In biological and medical sciences.

What is the quotient rule for square roots

To find the quotient rule for square root, let's take an example which is given as:

Consider h(x) = √x^2 + 1 / √x + 2

Solution:

Find the derivative of u(x) and v(x) such as:

$$ u(x) \;=\; (x^2 + 1)^{\frac{1}{2}} $$

$$ u’(x) \;=\; \frac{1}{2}(x^2 + 1)^{-\frac{1}{2}} . 2x \;=\; \frac{x}{\sqrt{x^2 + 1}} $$

$$ v(x) \;=\; (x + 1)^{\frac{1}{2}} $$

$$ v’(x) \;=\; \frac{1}{2} (x + 1)^{-\frac{1}{2}} . 1 \;=\; \frac{1}{2\sqrt{x + 2}} $$

Put these values in the quotient rule formula,

$$ h’(x) \;=\; (\frac{x}{\sqrt{x^2 + 1}}) . \sqrt{x + 2} - (\sqrt{x^2 + 1} . (\frac{1}{\frac{2\sqrt{x + 2)}}{(x + 2)}} $$

$$ =\; \frac{\frac{x \sqrt{x + 2}}{\sqrt{x^2 + 1}} - \frac{\sqrt{x^2 + 1}}{2\sqrt{x + 2}}}{x + 2} $$

$$ =\; \frac{\frac{2x \sqrt{x + 2} - \sqrt{x^2 + 1}}{2\sqrt{x + 2} \sqrt{x^2+1}}}{x+2} $$

$$ =\; \frac{2x \sqrt{x + 2} - \sqrt{x^2 + 1}}{2 \sqrt{x + 2} \sqrt{x + 2} \sqrt{x^2 + 1}} $$

Therefore the derivative solution of the given quotient rule question is,

$$ h’(x) \;=\; \frac{2x \sqrt{x + 2} - \sqrt{x^2 + 1}}{2(x + 2) \sqrt{x + 2} \sqrt{x^2+1}} $$

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Amanda Jepson

Last Updated February 21, 2024

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