The Chain Rule — Most Important Rule

Chain Rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)

"Derivative of the outside, times derivative of the inside."

Example: d/dx[sin(3x²)] = cos(3x²) · 6x

Outside: sin(□) → cos(□). Inside: 3x² → 6x.

Example: d/dx[(2x+1)⁵] = 5(2x+1)⁴ · 2 = 10(2x+1)⁴

Outside: □⁵ → 5□⁴. Inside: 2x+1 → 2.

Nested chain rule: d/dx[eˢⁱⁿ⁽ˣ²⁾] = eˢⁱⁿ⁽ˣ²⁾ · cos(x²) · 2x

Three layers: e^□ → e^□, sin(□) → cos(□), x² → 2x.

The chain rule appears in 80%+ of derivative problems. If you see a function inside another function, use the chain rule. Always ask: "Is there an inside function?"

Reference:

Wikipedia: Chain Rule

https://en.wikipedia.org/wiki/Chain_rule