I need to find a function F such that it is continuous everywhere and
and
The only thing i could think of is but that obviously doesnt satisfy the initial value. Any help or hints is greatly appreciated.
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.