This is an infinite power tower of x. You may have seen such a tower equated to a number, which is then solved to give an unexpectedly clean solution. The natural next step is (of course) to take the derivative.
There are a few too many xs here to use the chain rule, to say the least. If the power tower was, say, 10 tall, this would be excruciating but possible. The infinite height makes this problem a lot easier — notice that at each point in the tower, we still have an infinite tower from that point onwards. We can simplify the problem with the following substitution: