How to derive MRTS = 1/2 < 1 = w/r when production function equals √L+2K?

Question

I'm finding the total cost of a production function. How do I derive the marginal rate of technical substitution (MRTS) solution as 1/2 < 1 = w/r, and why is it less than (<) as opposed to = w/r?

The first step in calculating the total cost is calculating the MRTS. I am calculating the MRTS using the following formula:

MRTS formula

Therefore, I derived the MP of labor (L), substituting L for X.

MP of L

Then, I derived the MP of capital (K), substituting K for Y.

MP of K

Putting this into the original MRTS formula, I expect the answer to be 3/[2 sqrt (L+2K)] = w/r

Incorrect MRTS

the given solution is however 1/2 < 1 = w/r.

Correct MRTS