Let's throw out some junk and simplify the sentence as follows:

As the lines have become thinner and the circuits more complex, both the power of the chips and the devices they drive have increased.

We can use 2 splits to solve this quickly.

Split #1
"As the lines have become thinner and the circuits more complex" (answers A, B)
"As the lines have become thinner and circuits are more complex" (answers C, D)

A & B are superior because they maintain the verb tense: lines have become thinner; circuits (have become) more complex. In the 2nd part, the verb form is implied. Only change verb tense when you have a specific reason for doing so. Eliminate C and D.

Split #2
"both the power of the chips and the devices they drive have increased increased" (A, E)
"the power of both the chips and the devices they drive has increased" (B)

B is superior because what increases is the power, so A & E commit a subject-verb agreement error (the power have increased).

A & E can also be eliminated for idiomatic error: the construct "both X and Y" should have parallel X and Y.
Wrong: both the power of the chips and the devices they drive (A & E)
Right: the power of both the chips and the devices they drive (B)
Right: both the power of the chips and the power of the devices they drive

Answer B is correct.

PS. I didn't use this in my solution, but the split early between using a pronoun "their circuits" (C, D & E) and using a possessive noun "the chips' circuits" (A & B) can be used to get rid of C, D and E. Using the possessive noun is clearer because it leaves no room for doubts about whose circuits is being discussed (etched lines' or memory chips')

Hope this helps. My signature below has more info.
-Patrick