What does differentiable mean for a function?

Answer 1

geometrically, the function #f# is differentiable at #a# if it has a non-vertical tangent at the corresponding point on the graph, that is, at #(a,f(a))#. That means that the limit
#lim_{x\to a} (f(x)-f(a))/(x-a)# exists (i.e, is a finite number, which is the slope of this tangent line). When this limit exist, it is called derivative of #f# at #a# and denoted #f'(a)# or #(df)/dx (a)#. So a point where the function is not differentiable is a point where this limit does not exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different one-sided limits (a cusp, like for #f(x)=|x|# at 0). See definition of the derivative and derivative as a function.

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Answer 2

A function is considered differentiable at a point if it has a derivative at that point. Geometrically, this means that the function has a well-defined tangent line at that point, and the derivative represents the slope of that tangent line. Mathematically, a function ( f(x) ) is differentiable at a point ( x = a ) if the following limit exists:

[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} ]

If this limit exists, then the function is differentiable at ( x = a ). If a function is differentiable at every point in its domain, it is called differentiable over its entire domain.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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