As much as these doomsayers want for the earth to stop spinning, that won't happen with the traditional derivatives market.
Think of your state government. It could decide to go into the muni market and borrow a billion dollars. But they are worried that interest rates may go up between now and the time it takes to get the muni offering ready. So they call up JP Morgan and buy and interest rate cap. Let's say on a billion dollars the cap costs 100,0000 dollars. If rates go above the cap JP Morgan owes your state money.
Now, in this scenario, the notional amount of the cap (or derivative) is a billion dollars. But if JP Morgan goes out of business in the meantime your state is only out a hundred grand plus any extra interest rate expense caused by the loss of the cap.
And JP Morgan will turn around and hedge the cap, creating another billion in derivatives. (Which means they probably won't go out of business.)
And all these billion dollars are included when the worry warts talk about 400 trillion derivatives.
What you don't point out when you talk down to everyone is that it is not that $400T at risk. It is that small mistakes or market inefficiencies or unanticipated defaults could put 1% of that at risk and 1% is $4T which would go a long way towards tanking our economy and did, on a smaller scale, tank LTCM, requiring another "free to the public" Federal Reserve sponsored emergency rescue effort.