Calculus: Chain Rule Game/Worksheet

Calculus: Chain Rule Game/Worksheet
Master calculus with the Chain Rule Game. Master the chain rule for derivatives with Chain Reaction! Practice differentiating nested composite functions, exponential models, and trigonometric curves online. Include step-by-step solutions. Scroll down the page for more details.

 

 

How to Play The Game
Identify the Nested Functions:
Every round delivers a new composite equation F(x) = f(g(x)). Determine which part is the outer shell f(u) and which is the inner function g(x).

Locate the Evaluation Target:
Look at the bottom of the active game card to see your point of evaluation c (typically c = 0 or c = 1).

Peel Back the Layers:
Use the foundational chain rule mantra: “Derivative of the outside evaluated at the inside, times the derivative of the inside."

Input Your Scalar Answer:
Calculate the final integer value, type it into the input box, and hit Enter or click Verify Linked Velocity.

Learn from System Breakdowns:
If your chain breaks, the game generates a complete step-by-step breakdown using MathJax formatting to show you exactly where your math or substitution went astray.

The Underlying Mathematics
The mechanics of this game are powered entirely by the Chain Rule, which allows us to find the rate of change of composite functions.

The Mathematical Formula
When a function is nested within another, its derivative is defined as:

\(\frac{d}{dx}[f(g(x))] = f’(g(x)) \cdot g’(x)\)

Step-by-Step Game Mechanics Example
The engine dynamically cycles through three architectural formats: polynomial powers, exponentials, and trigonometric functions. Let’s look at how the game computes a Power Function problem:

Active Expression:
F(x) = (3x - 1)³ evaluated at c = 1

Step 1: Isolate the Inner and Outer Shells
Outer Layer: f(u) = u³ ⇒ f’(u) = 3u²
Inner Layer: g(x) = 3x - 1 ⇒ g’(x) = 3

Step 2: Combine via the Chain Rule
Following the formula, multiply the derivative of the outer wrapper by the derivative of the inner function:

F’(x) = 3(3x - 1)² · (3)
F’(x) = 9(3x - 1)²

Step 3: Substitute and Evaluate at c = 1
First, see what value the inner function yields at our target coordinate:

g(1) = 3(1) - 1 = 2

Now, complete the full calculation:

F’(1) = 9 · (2)²
F’(1) = 9 · 4 = 36

Entering 36 into the text box satisfies the evaluation condition, secures a 300-point increase, and increments your consecutive streak counter.

Chain Rule

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