This problem is a very simple example of a differential equation: an equation that relates a function to one or more of its derivatives. You can solve this problem by doing some educated guessing. ("educated" means "remember what we did in the past.")
Suppose $f$ is the function that satisfies $$f'(x) = -f^2(x)$$ for all $x$ in its domain, and $$f(1) = 1.$$ Then
$f(x) =$ .