E.3 Advanced Derivatives and Extrema
Advanced differentiation: the five standard derivatives (x^n, sin x, cos x, e^x, ln x) plus the three combination rules — chain for nested f(g(x)), product for u·v, and quotient for u/v. The SECOND derivative f''(x) reveals SHAPE: positive ⇒ concave up, negative ⇒ concave down, sign change ⇒ point of inflection. Optimisation locates maxima and minima by solving f'(x) = 0 and classifying via f''(x).
Standard derivatives and the three rules — chain, product, quotientSign up
Standard derivatives — the toolkit · Chain, product, quotient — three rules
Second derivative and graphical behaviourSign up
What is f''(x)? Rate-of-change of slope · Concavity and inflection — what f'' reveals
Maxima, minima, optimisation, and points of inflectionSign up
Finding maxima and minima · Optimisation — calculus in real life · Points of inflection