Derivative of U(X(t),t) with respect to t

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SUMMARY

The derivative of U(X(t),t) with respect to t is accurately represented as dU/dt or U'(X(t),t). According to the chain rule, the correct formulation is Ux(x,y)xt + Uy(x,y)yt, where Ux and Uy denote partial derivatives of U with respect to x and y, respectively. The expression Ut(Xt(t),t) is incorrect, as it implies differentiation with respect to both X(t) and t, which is not applicable in this context. The focus remains solely on the rate of change of U concerning time.

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What is the derivative of U(X(t),t)?

Is it Ut(Xt(t),t)?
 
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In general, the derivative of U(x,y) with respect to t is, by the chain rule,
Ux(x,y)xt+ Uy(x,y)yt.

Notice that the derivatives of x and y are multiplied by U(x,y), not arguments in it!

In the case that y= t, that reduces to
Ux(x,y)(xt)+ Ut.
 


The derivative of U(X(t),t) with respect to t is denoted as dU/dt or U'(X(t),t). It represents the rate of change of U with respect to time.

The expression Ut(Xt(t),t) is incorrect. It should be U'(X(t),t) or dU/dt. The notation Ut(Xt(t),t) suggests that the derivative is being taken with respect to both X(t) and t, which is not the case. The derivative of U(X(t),t) is only with respect to t.
 

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