🚀 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

Big changes don’t happen all at once—they grow from many tiny steps. Follow how breaking things into pieces reveals hidden structure.

🎥 2) Watch the Video

Watch the 3Blue1Brown video Derivative formulas through geometry | Chapter 3, Essence of calculus .

In this video, you’ll see how derivative rules can grow naturally out of simple geometric ideas rather than memorizing formulas. By exploring patterns in shapes and how they change, you’ll begin to understand why these rules work, not just how to use them. As you watch, look for the structure behind the formulas — this will help you transition from visual intuition to more algebraic and symbolic thinking.

It shows how formulas like d(x²)/dx = 2x can be seen as geometry.

✏️ 3) Do the Adventure Activity

✏️ Open: Adventure 3 — Activity

Build a big result by adding up lots of very small contributions.

🧠 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.