1. Spot the Layers: What is the "Inside" ($g$)? What is the "Outside" ($f$)?
2. Derivative of the Outside: Differentiate the shell, but keep the inside the same!
3. The Multiplier: Multiply your result by the derivative of the inside ($g'$).
4. Box Method: $(\text{Box})^n \to n(\text{Box})^{n-1} \cdot \text{Box}'$.
For each function, identify the Inner ($g$) and the Outer ($f$).
$y = (x^2 + 10)^5$
$y = \sin(3x)$
$y = \sqrt{5x - 2}$
Find the derivative ($y'$) for each function.
$y = (x^2 + 1)^3$
$y = (4x - 5)^{10}$
$y = \sin(x^2)$
If you have $y = (2x + 5)^1$... what is the derivative? Does the Chain Rule give you the same answer as the Power Rule? Explain.
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A machine has an inner gear ($u$) and an outer gear ($y$).
$y = u^2$ (The outer speed is the square of the inner).
$u = 5x + 1$ (The inner speed depends on the pedal $x$).
Find the rate of change of the whole machine ($\frac{dy}{dx}$).
Find the derivative of $y = \sin((x^2 + 1)^3)$.
This has three layers!
1. Sine (Outside)
2. Cube (Middle)
3. $(x^2 + 1)$ (Inside)
Objective: Explain the Chain Rule to a younger sibling using boxes.
The Activity:
1. Put a toy in a small box. Put the small box in a large box.
2. Tell them: "If I move the large box 1 foot, the toy moves 1 foot. But if I also move the small box *inside* the large one, the toy moves even more!"
The Lesson: "Total change is the Outer move plus the Inner move. In math, we multiply them to find the new speed."
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