Differentiate twice with respect to z:

1) Regard x and y as constants and use the chain rule:

u = z + x^3 + y^2

f(u)= u^4

f'(u)*u'= 4(z + x^3 + y^2)^3 * 1

f_z = 4(z + x^3 + y^2)^3

2) Chain rule again:

u = z + x^3 + y^2

f(u)= 4(u^4)

f'(u)*u'= 3*4(z + x^3 + y^2)^2 * 1

f_zz = 12(z + x^3 + y^2)^2