assuming you mean #(dT)/(dz)#, then by the product and chain rules:
#(dT)/(dz) = (d (4cot^3(4z)))/(dz)= 3*4 cot^2(4z) * d/dz(cot(4z))#
# = 12 cot^2(4z) * d/dz(cot(4z)) qquad triangle#
now #d/(dx) cot p = - csc^2 p #
if we have #p = 4 z# then by the Chain Rule
#d/(dz) cot p(z) = - csc^2 p (dp)/(dz) = = - csc^2 p * 4 = - 4 csc^2 z#
so #triangle# becomes
# = 12 cot^2(4z) * (- 4 csc^2 4z)#
# = -48 cot^2(4z) * csc^2 (4x)#
# = -48 (cos^2(4z))/(sin^2(4z)) * (1)/(sin^2 (4x))#
that simplification is going nowhere so we can stick with the previous answer
#T_z = -48 cot^2(4z) * csc^2 (4x)#
