Question

What is the second derivative of y= 8x^(1/3).
Possible answers are: 48
16
-32
32
-16
I am quite confused on this concept of derivative but this is what I have:
d^2/dx^2*f(x)= 8x^(1/3)
fx= d^2/dx^2 * 8x^(1/3)
fx= d/dx * (d/dx* 8x^(1/3)
fx= d/dx * (8*d/dx*x^(1/3)
fx= d/dx * (8*1/3*x^(1/3-1)
fx= d/dx * (8/3*x^(-2/3)
fx=8/3* d/dx * x^(-2/3)
fx=8/3*-2/3* x^(-2/3-1)
fx=8/3*-2/3* x^(-2/3-1)
fx=-16/9* x^(-5/3)
fx=-16/(9* x^(-5/3))
But how do I get from here to my answers??

Answer 1

y=8*x^(1/3).

First derivative:

y(') = 8 * (1/3) * x^(1/3 - 1) = (8/3) * x^(-2/3)

Second derivative:

y('')=(8/3)*(-2/3)*x^(-2/3 - 1)= (-16/9) * x^(-5/3).

Or, to put it in a different form

y('')= -16/ ( 9 * x ^ (5/3) )= -16 / (9* cuberoot (x^5) )

So, you're spot on with the derivative imo.
The only thing that's missing here is a value.
Are you sure there's nothing missing from the problem ? A context of some sort ? Or maybe a typo like x^3 instead of x^ (1/3).

Answer 2

"Find the second derivative" is the only context.