Final answer:
The solution involves calculating the derivatives (dx/dt) and (dx/ds) of an implicit function where x is dependent on t and s, examining different scenarios for these rates of change.
Step-by-step explanation:
The question involves implicit functions and how they change with respect to other variables or parameters. Specifically, it asks to find the derivatives (dx/dt) and (dx/ds) when x depends on t and s. We're looking for the rate of change of x with respect to time and the rate of change of x with respect to another variable, s.
When a function x(t, s) is implicitly defined in relation to variables t and s, finding the partial derivatives dx/dt and dx/ds involves holding the other variable constant while differentiating with respect to the one of interest. In this scenario where x only depends on t and s, it is implied by the provided statements that (dx/dt) and (dx/ds) can have different characteristics.
For instance, circumstances where these derivatives are independent of x and s, equal to zero, or infinite, change the behavior of the implicit function in response to changes in t and s.