Derivative (Quotient Rule) Game
The Quotient Rule is a method used to find the derivative of a function that is the ratio (or quotient) of two other differentiable functions. Scroll down the page for a more detailed explanation.
 
In this game, you will need to find the derivative of a function using the Quotient Rule and other derivative rules. Select one of the four possible answers. The correct answer will be highlighted in green. It includes a scoring system, and clear feedback to help you master this skill.
 

Quotient Rule Challenge

Progress Score: 0

Find the Derivative

Next Problem ➜

 

How to Play the Derivative (Quotient Rule) Game
The game will show you a function. Your task is to differentiate the function and select the correct answer.
Here’s how to play:

1. Start: Each Quiz consists of 10 questions. Click “Start Game”.
   
2. Look at the Problem: You will be given a function.
   
3. Select Your Answer: Select the correct derivative.br>
4. Check Your Work: Click “Submit Selection”. The game will tell you if you’re correct. If you are wrong, the correct answer will be highlighted in green.
   
5. Get a New Problem: Click “Next Question” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game When you have completed 10 questions, click “Finish Game” to get your final score.
    
   

The Quotient Rule
The Quotient Rule is a method used to find the derivative of a function that is the ratio (or quotient) of two other differentiable functions.
The Formula
If you have a function \(f(x)\) defined as the ratio of two functions, \(u(x)\) (the numerator, or “high”) and \(v(x)\) (the denominator, or “low”), then its derivative \(f’(x)\) is given by:
\(f(x) = \frac{u(x)}{v(x)}\)
\(f’(x) = \frac{v(x)u’(x) - u(x)v’(x)}{[v(x)]^2}\)
A common mnemonic to remember the formula is: “Low dee-High minus High dee-Low, over Low squared."
Low: The denominator, \(v(x)\).
dee-High: The derivative of the numerator, \(u’(x)\).
High: The numerator, \(u(x)\).
dee-Low: The derivative of the denominator, \(v’(x)\).
Step-by-Step Procedure
To find the derivative of \(f(x) = \frac{u(x)}{v(x)}\):
Step

1. Identify
   Identify \(u(x)\) (high) and \(v(x)\) (low).
   
2. Differentiate
   Find the derivatives \(u’(x)\) and \(v’(x)\).
   
3. Substitute
   Substitute all four components into the formula: \(\frac{v u’ - u v’}{v^2}\).
   
4. Simplify
   Simplify the numerator to get the final derivative.
    
   Detailed Example
   Problem: Find the derivative of \(f(x) = \frac{3x - 5}{2x^2 + 1}\).
   Step 1: Identify u and v
   \(u(x) = 3x - 5\) (High)
   \(v(x) = 2x^2 + 1\) (Low)
   Step 2: Find the Derivatives (u’ and v’)
   \(u’(x) = 3\) (dee-High)
   \(v’(x) = 4x\) (dee-Low)
   Step 3: Apply the Quotient Rule Formula
   \(f’(x) = \frac{\overbrace{(2x^2 + 1)}^{v} \overbrace{(3)}^{u’} - \overbrace{(3x - 5)}^{u} \overbrace{(4x)}^{v’}}{\underbrace{(2x^2 + 1)^2}_{v^2}}\)
   Step 4: Simplify the Result
   First, expand the numerator:
   \(\text{Numerator} = (6x^2 + 3) - (12x^2 - 20x)\)
   Distribute the subtraction sign:
   \(\text{Numerator} = 6x^2 + 3 - 12x^2 + 20x\)
   Combine like terms:
   \(\text{Numerator} = -6x^2 + 20x + 3\)
   Final Derivative:
   \(f’(x) = \frac{-6x^2 + 20x + 3}{(2x^2 + 1)^2}\)
    
   

This video gives a clear, step-by-step approach to learn how to find the derivative of a function using the quotient rule.

• Show Video

 

Try out our new and fun Fraction Concoction Game.

Add and subtract fractions to make exciting fraction concoctions following a recipe. There are four levels of difficulty: Easy, medium, hard and insane. Practice the basics of fraction addition and subtraction or challenge yourself with the insane level.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.