Limit of derivative of implicit function

Question

Define $f(x)=x(1-x)^s$, where $x\leq 1$ and $s>0$. Note that this is an inverted-U-shaped function with peak at $x=1/(1+s)$. Given $x$ not equal to $1/(1+s)$, define $y(x)$ implicitly by (i) $f(y)=f(x)$ and (ii) $y\neq x$. That is, $y(x)$ is the other value of the argument at which $f$ attains value $f(x)$. I would like to compute the limit of $y'(x)=f'(x)/f'(y(x))$ as $x$ goes to $1/(1+s)$ from below. I’ve experimented around numerically and I’m pretty sure this limit equals $-1$, but I can’t prove it.