Ryan Collison

Why does differentiating sine give cosine, and cosine give negative sine? The answer lives on a spinning wheel: position and velocity are exactly 90 degrees out of phase, and that geometry is the proof.

Every other function changes shape when you differentiate it. eˣ stays the same. That fixed-point property isn't a coincidence — it's the definition of e, and it's why this number shows up in everything from compound interest to quantum mechanics.

A ladder slides down a wall. As the base moves outward, the top drops — but not at the same speed. Related rates problems capture these coupled changes using the chain rule, turning geometric or physical constraints into precise equations about rates.

Some curves are defined by equations where y can't easily be solved for. Implicit differentiation finds the derivative anyway — differentiating both sides with respect to x and treating y as a hidden function of x.

When you differentiate a ratio of two functions, the denominator's rate of change subtracts from the numerator's — scaled by g squared. The quotient rule captures this precisely, and understanding where it comes from (it's just the product rule in disguise) makes it impossible to misremember.

Students often guess the derivative of f times g is f' times g'. It's not. The product rule — f'g + fg' — captures something subtler: both functions are changing, and their interaction creates a cross term that naïve reasoning misses. The geometric proof is the cleanest explanation.