What you'll learn
Differentiating more complex functions: composites (chain rule), implicit relations, and inverses.
Chain rule
For y = f(g(x)): dy/dx = f′(g(x))·g′(x) — derivative of the outside (leaving the inside) times derivative of the inside.
- Example: d/dx[sin(3x²)] = cos(3x²)·6x.
Implicit differentiation
When y isn't isolated, differentiate both sides with respect to x, treating y as a function of x (so d/dx[y²] = 2y·dy/dx), then solve for dy/dx.
- Example: x² + y² = 25 → 2x + 2y·y′ = 0 → y′ = −x/y.
Inverse functions
(f⁻¹)′(x) = 1 / f′(f⁻¹(x)). Useful derivatives: d/dx[ln x] = 1/x, d/dx[arcsin x] = 1/√(1−x²), d/dx[arctan x] = 1/(1+x²).
Exam tips
- Identify inside vs outside before applying the chain rule.
- After implicit differentiation, isolate y′.
Common mistakes
- Forgetting the ·g′(x) factor in the chain rule.
- Dropping dy/dx when differentiating y-terms implicitly.