If a derivative of f(x) is zero at a specific point is it still differentiable - Calculus 1?

Question

So I've been thinking about this for the past hour or so, I asked my friend in Calc3 and he wasn't sure so here I am.

If the derivative of f(x) is 0 at a specific point can we still say that the function is differentiable at that point just that no differentiation is occurring?

But if no change is occurring in one variable with respect to another, doesn't that mean it's non-differentiable?

This seems paradoxical because the slope is defined as 0 but there is no change occurring.

Any input here would be great :)

Answer 1

The local max and mins of a function have a zero value for the derivative at these points and are differentiable.

A function is differentiable at a point if it is continuous at that point, and the left sided derivative is equal to the right sided derivative (this excludes cusps and corners). The value of the derivative is not a requirement.

Answer 2

I think you have two competing definitions of "differentiation" in your head. One of them is "compute the slope at a particular point", which is essentially correct. The other is something vague about change in one variable with respect to another. Your "paradox" is just caused by your second "definition" being ill-defined and/or wrong.



So, try to make the definition of a derivative very precise, and see if that clears up your confusion.



(P.S. Your friend in Calc 3 should have been able to tell you immediately that you can have zero derivative at a point, since by then you will definitely have done min/max finding, and the whole point is to set the derivative equal to zero.)

Answer 3

If the derivative of f(x) is 0 at a specific point can we still say that the function is differentiable at that point just that no differentiation is occurring?

- yes , it's differentiable at this point



non-differentiable means we do not know the change

Answer 4

If the derivative is zero, it means the rate of change is zero; not that the function is non-differentiable. The derivative is zero wherever your function changes direction (e.g., the vertex of a parabola).



So, to summarize, the function is differentiable at that point, but the rate of change is zero.

Answer 5

If it was not differentiable at the point then derivative would not have any value not even 0 at that point

Answer 6

look up the definition of " differentiability " and answer your query ...