Derivative (Chain Rule) Game
The Chain Rule is used to find the derivative of a composite function — a function inside another function. To use the Chain Rule, you first identify the “outer” and “inner” functions, then differentiate the outer function while treating the inner function as a single variable, and finally multiply that result by the derivative of the inner function. Scroll down the page for a more detailed explanation.
 
In this game, you will need to find the derivative of a function using the Chain 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.
 

Chain Rule Challenge

Progress Score: 0

Find the Derivative

Next Problem ➜

 

How to Play the Derivative (Chain 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 Chain Rule
The Chain Rule is used to find the derivative of a composite function—a function inside another function.
The Rule
If you have a function \(y = f(g(x))\), where \(f\) is the “outside” function and \(g\) is the “inside” function, the derivative is:
\(\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)\)
In simpler terms (The “Outside-Inside” Rule):
Differentiate the outside function (keep the inside exactly the same).
Multiply by the derivative of the inside function.

Step-by-Step Example (Power Rule)
Problem: Find the derivative of \(y = (3x^2 + 1)^5\).
Step 1: Identify Outside and Inside
Think of this as layers:
Inside (\(u\)): \(3x^2 + 1\)
Outside (\(f(u)\)): \(u^5\) (something to the power of 5)
Step 2: Differentiate the Outside
Bring the power down and subtract 1 (standard Power Rule), but leave the inside alone.
Outside derivative: \(5(\dots)^4\)
Result so far: \(5(3x^2 + 1)^4\)
Step 3: Differentiate the Inside
Now look only at the inside part (\(3x^2 + 1\)).
Inside derivative: \(\frac{d}{dx}(3x^2 + 1) = 6x\)
Step 4: Multiply Them Together
\(y’ = \underbrace{5(3x^2 + 1)^4}_{\text{deriv of outside}} \cdot \underbrace{(6x)}_{\text{deriv of inside}}\)
Simplify:
\(y’ = 30x(3x^2 + 1)^4\)
 

Summary Checklist
When you see a complex function, ask yourself:

1. Is there an “inside” part? (Usually inside parentheses, under a square root, or inside a trig function).
   
2. Derivative of Outside: Apply rules (Power, Trig, etc.) to the outer shell.
   
3. Copy the Inside: Don’t change the inner function yet.
   
4. Multiply by Derivative of Inside: Chain it on at the end.
    
   

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

• Show Video

 

Derivative Challenge (Polynomials)
Find the derivatives of polynomials using the Power, Sum, Difference, and Constant Multiple rules.

Derivative Challenge (Binomials)
Find the derivatives of the product two binomial functions. Either use the Product Rule or multiply the two binomials first to get a single polynomial and use the Power Rule.

Derivative Challenge (Product Rule)
Find the derivatives of the product a polynomial and a trig function. Use the Product Rule.

Derivative Challenge (Quotient Rule)
Find the derivatives of a function using the Quotient Rule.

Derivative Challenge (Chain Rule)
Find the derivatives of a function using the Chain Rule.

Derivative Rules
Find the derivatives of a function using the Power Rule, Constant Rule, Sum/Difference Rule, Product Rule, Quotient Rule, Chain Rule, and the rules for trigonometric and exponential functions.

Derivative Challenge (Trig Function)
Find the derivatives of Trigonometric Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Exp, Log)
Find the derivatives of Exponential and Logarithmic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

Derivative Challenge (Hyperbolic)
Find the derivatives of Hyperbolic Functions. May require other basic rules for example the Chain Rule, Product Rule etc.

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.