Lesson 7: Chain Rule – The Basics & Intuition

1. Why the Chain Rule?

Many functions are compositions: one function inside another.

Example: \(y = (3x + 2)^5\)

• Outer function: \(u^5\) (where \(u = 3x + 2)\)
• Inner function: \(u = 3x + 2\)

The chain rule tells us how to differentiate compositions.

2. Chain Rule Formula

If \(y = f(g(x))\), then:

\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

Memory aid: "Derivative of the outside, times derivative of the inside"

Example: \(y = (3x + 2)^5\)

Let \(u = 3x + 2 → y = u^5\)

\(dy/du = 5u^4,\, du/dx = 3\)

\(dy/dx = 5(3x + 2)^4 · 3 = 15(3x + 2)^4\)

3. Chain Rule Intuition

Think of it as multiplying rates:

• How fast is the inside changing? → \(g'(x)\)
• How fast is the outside changing with respect to the inside? → \(f'(g(x))\)

Together: overall rate of change.