Composite, Implicit & Inverse

🔗 chain rule

In Leibniz notation: if y = f(u) and u = g(x), then dy/dx = dy/du · du/dx.

For nested chains, multiply derivatives layer by layer (like peeling an onion 🧅).

🌙 implicit differentiation

Use when x and y are mixed and solving for y is annoying. Differentiate both sides with respect to x, remembering d/dx[y] = y′ (or dy/dx) and chain rule on yTerms.

↩️ inverse functions

For inverse trig, memorize the shapes (constants over square roots).

📐 inverse trig derivatives

• d/dx [arcsin x] = 1 / √(1 − x²)
• d/dx [arccos x] = −1 / √(1 − x²)
• d/dx [arctan x] = 1 / (1 + x²)

Don't forget chain rule multipliers when the inside isn't just x.

📈 higher-order derivatives

f″, f‴, f⁽⁴⁾… each is the derivative of the previous. Notation: d²y/dx² for second derivative.

Acceleration in motion: if s(t) is position, v = s′, a = s″.