Application of Derivatives

\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}, \text{ if } \frac{dx}{dt} \neq 0

\frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r

V = x^3, S = 6x^2

P = 2(x + y), A = x \cdot y

A function f is increasing on I if x1 < x2 in I implies f(x1) \le f(x2).

A function f is decreasing on I if x1 < x2 in I implies f(x1) \ge f(x2).

A function f is constant on I if f(x) = c for all x in I.

A function f is strictly increasing on I if x1 < x2 implies f(x1) < f(x2).

A function f is strictly decreasing on I if x1 < x2 implies f(x1) > f(x2).

If f has a local maxima or a local minima at x=c, then either f'(c)=0 or f is not differentiable at c.

A point c in the domain of f at which either f'(c)=0 or f is not differentiable is called a critical point of f.

First Derivative Test: f'(x) > 0 left of c, f'(x) < 0 right of c implies c is local maxima.

First Derivative Test: f'(x) < 0 left of c, f'(x) > 0 right of c implies c is local minima.

Second Derivative Test: f'(c)=0, f''(c)<0 implies local maxima; f''(c)>0 implies local minima; f''(c)=0 means test fails.

If f is continuous on [a,b], then f has an absolute maximum and an absolute minimum value, each attained at least once in [a,b].

If f is differentiable on closed interval I and c is an interior point where f attains absolute max or min, then f'(c)=0.