Question

Is the function x^(1/3) differentiable at 0? why or why not?

Answer 1

try to find the derivative function of x^(1/3)

Answer 2

it is continuous, so yes

Answer 3

but is the function smooth there?

Answer 4

The function needs to be continuous and smooth at x=a to be differentiatiable at x=a

Answer 5

derivative of x^1/3= 1/(3x^(2/3))

Answer 6

Now how do I prove that it's differentiable?

Answer 7

if it is continuous so yes? this is much easier than that
take the derivative
if you can evaluate it at zero, then yes
if you cannot, then no

Answer 8

smooth... like gary coleman

Answer 9

lol

Answer 10

you may use limit definition to make your proof a bit more lengthy

Answer 11

@aawowaa you can't divide by 0

Answer 12

But if you are looking for something more formal
there is that thingy that @ganeshie8 mentioned

Answer 13

derivative = 1/0 at a point tells us that the tangent line is vertical

Answer 14

So the function is not differentiable... right?

Answer 15

derivative of x^1/3= `1/(3x^(2/3))`



show that this value is not defined at x = 0

Answer 16

1/(3(0)^(2/3))=1/0 so it's undefined

Answer 17

Correct. Differentiable means the derivative exists.
At x = 0, the derivative is undefined.
Therefore, at x = 0, the function is not differentiable.