It's . Essentially, you treat g(x) as if it were a variable for the first part, then multiply by the derivative of g. For your problem, you have to do the chain rule twice, since you have f(g(a third function)).
For the second one, I think you just calculate the derivative and look at the formula to see where it's undefined (no 0 in denominator, no square root of negative, no ln of negative or zero).
For the third, I think your analysis is correct. You can combine the logarithms to get it to be defined, but as it's written the function is undefined everywhere.
For the fourth, try calculating the distance after 1 hour.
And for the last, you can find dS/dr. Use this as an estimate of (error in S)/(error in r). But the % error in S means (error in S)/S. So