Find the rate of change of the volume of a cube

Find the rate of change of the volume V of a cube with respect to the length of w of a diagonal on one of the faces at w=6.

I know how to do this when given a side but have never done this with a diagonal. i tried solving this way
i solved for the side of one of the faces which is 2sqrt(3). i then took the derivative of V=s^3
which is V'=3s^2 i then plugged in the length of the side and came up with 36 as my answer. Im not sure if i did this correctly because it is not one of the answers given on my homework and im very wary of choosing "none of the above". Can someone help me please?
The other answer choices are
a)27
b)27sqrt(2)
c)27sqrt(2)/2
d)36sqrt(2)

V = a^3
w = asqrt(2)
a = w/sqrt(2)
V = w^3 /(2sqrt(2))
dV/dw = 3w^2/(2sqrt(2))
when w = 6
dV/dw = 54/sqrt(2) = 27 sqrt(2)