🚀 Adventure 3 — Free Fall

Grow a square and a cube by a tiny amount dx and watch the Power Rule appear.

🎯 What you will learn

If f(x)=x² then f'(x)=2x.

If f(x)=x³ then f'(x)=3x².

The “extra tiny pieces” involve dx² and dx³, and they vanish as dx → 0.

📖 1) Read / Listen: Story

📖 The Apple, the Bet, and the Book

🎥 2) Watch the Video

Watch the 3Blue1Brown video Derivative formulas through geometry | Chapter 3, Essence of calculus . It shows how formulas like d(x²)/dx = 2x can be seen as geometry.

✏️ 3) Do the Adventure Activity

✏️ Open: Adventure 3 — Activity

🧠 4) Questions to Keep in Mind

In the square picture, why are there exactly two strips of area x·dx?
In the cube picture, where do the 3 big sheets of volume x²·dx come from?
Why do we ignore the terms with dx² and dx³ when we take a derivative?

💡 5) If You Get Stuck

First: write df(x) as “(new area/volume) − (old area/volume)”.
Then: break the new pieces into types (strips, sheets, edge bars, corner cube).
Finally: divide by dx and let dx → 0.