# Differentiability

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

*x*_{0}is an interior point in the domain of a function*f*, then*f*is said to be*differentiable at x*_{0}if the derivative*f*â€˛(*x*_{0}) exists. This means that the graph of*f*has a non-vertical tangent line at the point (*x*_{0},*f*(*x*_{0})).The function

*f*may also be called*locally linear*at*x*_{0}, as it can be well approximated by a linear function near this point.

## Essential Questions

Essential Questions

### How is differentiability defined?

How is differentiability defined?

Linearization GIF.gif