Remember to keep the symbols straight; there is no y in the actual problem. Though I know what you mean. Acceleration is defined to be the second derivative of position with respect to time t, so we want to find s''(t).
First we find s'(t), which can be found by applying the chain rule, where the square root is the outer function and is the inner function. To keep things clear, I like to use exponents in place of roots, so I would write it out:
Explicitly, we can see that s(t) = f(g(t)) where and . By the chain rule, s'(t) = f'(g(t))g'(t).
In order to get the second derivative you will have to use the product rule in addition to the chain rule. Can you find s''(t) from here?