How would I go about showing the existence of a differentiable funciton over the reals such that, for all x,
?
You don't. You can't. It's not true. If there were a differentiable function satisfying that equation, then, differentiating both sides, . So either f'(x)= 0, for all x, or 5f(x)^4+ 1= 0. If f'(x)= 0 for all x, then f(x) is a constant, c, such that or which is impossible: the right side changes as x changes and the left side is a constant. However, all terms in 5f(x)^4+ 1 are non-negative: their sum is 0 only if 5f(x)^4= -1 which again says that f(x) is a constant.