Differentiable Function

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.
  • …
• 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.
  • …
• Counter-Example(s):
  • a Non-Differentiable Function.
  • a Continuous Function.
• See: Differential Equation, Well-Behaved Function, Delta Rule.

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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 x₀ is a point in the domain of a function ƒ, then ƒ is said to be differentiable at x₀ if the derivative ƒ′(x₀) is defined. This means that the graph of ƒ has a non-vertical tangent line at the point (x₀, ƒ(x₀)), and therefore cannot have a break, bend, or cusp at this point.

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