Derivative of the Solution to a Fixed Point Iteration

Question

Let $\theta_s$ is the solution to a fixed point equation $$\theta=f(\theta,\lambda)$$ Let $d(\theta,\lambda)$ be another function of $\theta,\ \lambda$. I know $f$ but I have no explicit expression for the solution $\theta_s$. Now, if I need to evaluate $d'(\theta_s,\lambda_s)$ where the differentiation is with respect to $\lambda$, how should I proceed to do that?

Answer 1

Use the Implicit Function Theorem. Your function $$\lambda\longmapsto\theta_s$$ is defined implicitly by $$\theta=f(\theta,\lambda).$$ Then, $$ \theta'(\lambda)= {\partial f\over\partial\theta}\,\theta'(\lambda)+{\partial f\over\partial\lambda}. $$