Use the Chain Rule to find dzds and dzdt.

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so I tried using the chain rule for dz/ds and started with first d second + second d first and used the chain rule by plugging in the value of r and 0 when taking the derivatives but my answer is completely different than the books answer... can someone show me the steps real detailed explaining each step so I can just make sure I'm doing this the right way again? After reading my notes I thought I was supposed to do the multiplication rule combined with the chain rule but it's coming out differently..