Editing Differentiable function (section)

{{short description|Mathematical function whose derivative exists}}
[[File:Polynomialdeg3.svg|thumb|right|A differentiable function]]
In [[mathematics]], a '''differentiable function''' of one [[Real number|real]] variable is a [[Function (mathematics)|function]] whose [[derivative]] exists at each point in its [[Domain of a function|domain]]. In other words, the [[Graph of a function|graph]] of a differentiable function has a non-[[Vertical tangent|vertical]] [[tangent line]] at each interior point in its domain. A differentiable function is [[Smoothness|smooth]] (the function is locally well approximated as a [[linear function]] at each interior point) and does not contain any break, angle<!--Please, do not link to [[angle]] as this is the common language meaning. A link to [[curvilinear angle]] would be possible if (or when) such an article would (or will) exist. -->, or [[Cusp (singularity)|cusp]].

If {{math|''x''<sub>0</sub>}} is an interior point in the domain of a function {{mvar|f}}, then {{mvar|f}} is said to be ''differentiable at'' {{math|''x''<sub>0</sub>}} if the derivative <math>f'(x_0)</math> exists. In other words, the graph of {{mvar|f}} has a non-vertical tangent line at the point {{math|(''x''<sub>0</sub>, ''f''(''x''<sub>0</sub>))}}. {{mvar|f}} is said to be differentiable on {{mvar|U}} if it is differentiable at every point of {{mvar|U}}. {{mvar|f}} is said to be ''continuously differentiable'' if its derivative is also a continuous function over the domain of the function <math display="inline">f</math>. Generally speaking, {{mvar|f}} is said to be of class {{em|<math>C^k</math>}} if its first <math>k</math> derivatives <math display="inline">f^{\prime}(x), f^{\prime\prime}(x), \ldots, f^{(k)}(x)</math> exist and are continuous over the domain of the function <math display="inline">f</math>.

For a multivariable function, as shown [[#Differentiability in higher dimensions|here]], the differentiability of it is something more complex than the existence of the partial derivatives of it.