Additional Theorems and Rules of Differentiable Functions

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If f(x) is differentiable at c, it is continuous at c.

The Extreme Value Theorem
If f(x) is continuous on the closed interval [a, b], then f has both a minimum and maximum on the interval.

Relative Extrema Occur only at Critical Numbers
If f has a relative minimum or relative maximum at x = c, then c is a critical number of f.

Rolle's Theorem
Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) such that f'(c) = 0.

Absolute Value Rule
If u is a differentiable function of x such that u does not = 0, then d/dx[ln | u |] = u'/u