Unit 3 · Differentiation: Composite, Implicit, and Inverse Functions

Chain Rule ● (core concept) — For a composite function f(g(x)), the derivative is f'(g(x))·g'(x) — the derivative of the outside times the derivative of the inside.

Composite function ○ — A function built by applying one function to the output of another, written f(g(x)) or (f ∘ g)(x).

Implicit differentiation ● (core concept) — Differentiating both sides of an equation relating x and y while treating y as a function of x (so the Chain Rule produces dy/dx terms), then solving for dy/dx.

Derivative of an inverse function ● (core concept) — If f is invertible and f(a) = b, then (f⁻¹)'(b) = 1 / f'(a), provided f'(a) ≠ 0.

Derivative of arcsin x ● (core concept) — d/dx[arcsin x] = 1/√(1 − x²).

Derivative of arctan x ● (core concept) — d/dx[arctan x] = 1/(1 + x²).

Derivative of arccos x ● (core concept) — d/dx[arccos x] = −1/√(1 − x²).

Second derivative ● (core concept) — The derivative of the derivative, written f''(x), y'', or d²y/dx²; it measures how the rate of change itself is changing.

Higher-order derivatives ● (core concept) — Derivatives beyond the second, denoted f'''(x), f⁽⁴⁾(x), …, f⁽ⁿ⁾(x).