Say I have a function $f(x,\theta)$ such that for each value of $\theta$ the function $f$ has a unique fixed point $x^*(\theta)$, i.e. $$x^{*}(\theta) = f(x^{*}(\theta),\theta)$$ My question is when can we conclude that $x^{*}$ is a differentiable function of $\theta$?
What I was able to show so far:
Say there are some constants $L < 1$ and $K \geq 0 $ so that 1) for all $\theta$ the function of $x$ obtained from $f$ by holding $\theta$ fixed is Lipschitz with modulus $L$ and 2) for all $x$ the function of $\theta$ obtained from $f$ by fixing $x$ is Lipschitz with modulus K. Then $x^*$ is Lipschitz with modulus $\frac{K}{1 - L}$ But I am trying to find a stronger condition now, i.e. that $x^*$ is differentiable.
I assume you mean you want a condition like that $f$ is differentiable. This condition is too strong, though. Consider $f(x,\theta) = x^3+x-\theta$. For each $\theta$, there is a unique fixed point, where $x = \sqrt[3]{\theta}$, but this is not differentiable w.r.t. $\theta$.