• In calculus, a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

  • As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each interior point in its domain, be relatively smooth, and cannot contain any break, angle, or cusp.

  • More generally, if x0 is an interior point in the domain of a function f, then f is said to be differentiable at x0 if the derivative f ′(x0) exists. This means that the graph of f has a non-vertical tangent line at the point (x0, f(x0)).

  • The function f may also be called locally linear at x0, as it can be well approximated by a linear function near this point.

Essential Questions

How is differentiability defined?

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