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