So you're supposed to find both F and y(t) for which the given conditions are true, or ...?
How about . Then .
The problem with this question is that the complexity of any derivative is arbitrary, for example, if we know that the function y is of the general form:
... where a and b are constants. It is easy to create a function F that will reliably produce y's derivative, as in:
However, even a slight change in the general form of y will upset everything and F will no longer function, while, as stated earlier the above function F produces the derivative of y, it will not produce the derivative of x (with constants a, b and c):
However, a bit of manipulation will create a function that will work for both x and y, (above) function G, since they are quite similar. this won't work in all instances; consider the functions p and q, defined below:
Creating a function that will produce the derivatives of both these functions is an enormous, if not impossible, task. So, my conclusion is, it is possible to create a function F that will produce the derivative of a function y, but the general form of y must first be known. Otherwise we'd have to consider an infinite number of combinations of roots, fractions, logarithms, trigonometric functions and lord knows what else - which from my humble high-school perspective - seems impossible.
Sorry. =( I hope someone else has a more positive answer.