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While working on a problem in plasma dynamics, I keep having terms of these forms come up $$ \frac{\partial}{\partial t}\left(\frac{dU}{dt}\right), \frac{\partial}{\partial U}\left(\frac{dU}{dt}\right) $$ where U and t are the two independent variables. My instincts tell me these are zero, but I'm having trouble showing it. Any help?

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    Could you give an example from the book? I infere from your question that you probably are missunderstanding something there. Be more explicit please, and give some context. – 2017-02-14

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I hope this helps you:

If $U$ doesn't depend on $t$, then the value of $U$ doesn't change when varying $t$, then $U$ is constant with respect to $t$, then $\dfrac{dU}{dt}$ represents the derivative of a constant, which it is always $0$.