Differentiability
Text According to the definition, a function f of one variable is said to be differentiable at a point x_{0} in its domain if there exists a function f '(x) such that
f'(x_{0}) = lim_{h > 0} [f(x_{0} + h)  f(x_{0})]/h
where h is a real number. This is equivalent to
lim_{h > 0} [f(x_{0} + h)  f(x_{0})  f'(x_0)*h]/h = 0.
This condition can be written in terms of epsilons and deltas as well. We say that f is differentiable at x_{0} if for any ε > 0 there exists a &delta > 0 such that
[f(x_0 + h)  f(x_0)  f'(x_0)h]/h < ε
or equivalently,
[ f '(x_{0}  ε) h < f(x_{0} + h)  f(x_{0}) < (f '(x_{0}) + ε) h
whenever h < &delta. This definition is similar in form to the definition of continuity. Recall that in the geometrical interpretation of continuity the challenge was to find a small enough &deltadisc domain of x_{0} such that the graph of the function over the domain would lie between two horizontal bars a distance ε above and below f(x_{0}). In the geometrical interpretation of differentiability, as seen in the inequality above, the challenge is to find a &deltadisc domain, centered at x_{0}, that is small enough so that the graph of f over the domain lies between lines with slopes (f'(x_{0}) + ε and (f'(x_{0})  ε).
Demos
Differentiability
 
This demo shows two blue lines through a point (x_{0}, f(x_{0})), one with a slope ε greater than that of the tangent line and one with a slope ε less than that of the tangent line. If the function is differentiable at x_{0}, you should always be able to choose a &delta (controlled by the hotspot) small enough that the portion of the curve between x_{0}  &delta and x_{0} + &delta lies inside the blue lines.

Exercises 1. Use the demo to test whether the following functions are differentiable at the points given:
 f(x) = x^2 at x = 5
 f(x) = tan(x) at x = π/4
 f(x) = x at x = 0
 f(x) = x at x = 1
2. Use the demonstrations for continuity and differentiability in one variable to give an argument for why differentiability at a point implies continuity at a point.
