Hello please can someone work out this complex calculus problem for me.

Question

A rectangular box has sides of length x, y and z. Write down an expression for the length of the body diagonal L in terms of x, y and z. If sides x and y are increasing by 2 cm/s and side z is decreasing by 3 cm/s, what is the rate of change of L when x = 2 cm, y = 3 cm and z = 3 cm.

Answer 1

L^2 = x^2 +y^2 + z^2 (a 3 dimensional version of the Pythagorean Theorem)

take derivatives with respect to time

2LL' = 2xx' + 2yy' + 2zz'

divide by 2

LL' = xx' + yy' + zz' plug in the values x'=y' =2, z' =-3, x=2, y=3, z=3, and L=sqr(4+9+9) =sqr22, find L'

L' = (2(2) + 3(2) + 3(-3))/sqr22 = (4+6-9)/sqr22 = 1/sqr22

L' = dL/dt = 1/sqr22 = sqr22)/22

L' = (22)sqr22 = about 0.213 cm/sec