Right derivative: f'+(a) = lim(h->0+) [f(a+h) - f(a)]/h Left derivative: f'-(a) = lim(h->0-) [f(a+h) - f(a)]/h

f is differentiable at a iff f'+(a) = f'-(a) = finite value.

At endpoints of intervals: Only one-sided derivatives make sense. f is differentiable at the left endpoint a of [a,b] if f'+(a) exists.

Example: f(x) = sqrt(x) on [0, infinity). f'+(0) = lim sqrt$\frac{h}{h}$ = lim 1/sqrt(h) = infinity. So f is NOT differentiable at 0 (infinite derivative).

Example: f(x) = x^$\frac{4}{3}$. f'(0) = lim h^$\frac{4/3}{h}$ = lim h^$\frac{1}{3}$ = 0. Differentiable at 0 with f'(0) = 0.