Please help with an Optimization derivative problem

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 8 km east of the refinery. The cost of laying pipe is $400,000/km over land to a point P on the north bank and $800,000/km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Round your answer to two decimal places.)

I am having a lot of difficulty with this problem.

I set up the cost function as : C = 400000x + 800000y
with y = sqrt( 4 + (8-x)^2 ) <--- found by pythag theorem

so the cost function would be : C = 400000x + 800000 * ( sqrt( 4 + (8-x)^2 )

but I'm having trouble getting the first derivative, and subsequently the critical value of x = ? which would be the answer.

Could someone please explain how to do this? I'd really appreciate it!

y^2 = 4 + (8 - x)^2 ... ⇒ dy/dx = (x - 8) / y
C = 4x + 8y

dC/dx = 4 + 8 dy/dx
= 4 + 8 (x - 8) / y
= 0 at y = 2 (8 - x)

(2(8 - x))^2 = 4 + (8 - x)^2
x = 2/3 (12 - √3)

Answer: 2/3 (12 - √3) ≈ 6.85 mi