Application of Derivatives

When one quantity y depends on

When one quantity y depends on another quantity x through y = f(x), the derivative dy/dx represents how fast y changes with respect to x. If both x an

A function f(x) is increasing on

A function f(x) is increasing on interval I if x₁ < x₂ implies f(x₁) < f(x₂). It's decreasing if x₁ < x₂ implies f(x₁) > f(x₂). Using derivatives: f'(

Points where f'(x) = 0

Points where f'(x) = 0 or f'(x) does not exist are called critical points. These are potential locations for local maxima, minima, or points of inflec

At a critical point c

At a critical point c: if f'(x) changes from positive to negative, c is a local maximum; if f'(x) changes from negative to positive, c is a local mini

For a critical point c where

For a critical point c where f'(c) = 0: if f''(c) < 0, then c is a local maximum; if f''(c) > 0, then c is a local minimum; if f''(c) = 0, the test is