Find the function that would produce a derivative of y = x. Prove your results.

Answer 1

y = 1/2 x^2
y' = 1/2 * (2x) = x

In fact,

y = x^2+c
has y' = x for any constant c.

For proof, are you looking for the derivative using the definition of the limit? If so, I'm sure you've seen the work before.