# Differentiable Function

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A Differentiable Function is a real-valued vector function that has a derivative function.

**Context:**- If [math]\displaystyle{ ƒ(x) }[/math] is Differentiable at [math]\displaystyle{ a }[/math], then
*ƒ*must also be Continuous at*a*. - It can have a Partial Derivative on one of its Dimensions.
- It can be the Input to a Function Differentiation Task.
- It can range from being a Continuously Differentiable Function to being a Non-Continuously Differentiable Function.
- …

- If [math]\displaystyle{ ƒ(x) }[/math] is Differentiable at [math]\displaystyle{ a }[/math], then
**Example(s):**- an Exponential Function
^{d}/_{dx}[math]\displaystyle{ f }[/math](x*) =*e^{x} - [math]\displaystyle{ f(x,y)=5xy^3 }[/math].
- a Differentiable at Zero Function.
- …

- an Exponential Function
**Counter-Example(s):****See:**Differential Equation, Well-Behaved Function, Delta Rule.

## References

### 2012

- (Wikipedia, 2012) ⇒ http://en.wikipedia.org/wiki/Differentiable_function
- QUOTE: In calculus (a branch of mathematics), a
**differentiable function**is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain. As a result, the graph of a differentiable function must be relatively smooth, and cannot contain any breaks, bends, or cusps, or any points with a vertical tangent.More generally, if

*x*_{0}is a point in the domain of a function ƒ, then ƒ is said to be*differentiable at*x_{0}if the derivative ƒ′(*x*_{0}) is defined. This means that the graph of ƒ has a non-vertical tangent line at the point (x_{0}, ƒ(*x*_{0})), and therefore cannot have a break, bend, or cusp at this point.

- QUOTE: In calculus (a branch of mathematics), a