Functional equation $f^{-1}(-x) = -f(x)$

Question

Is there a way to find all analytic functions $f:\mathbb{R} \to \mathbb{R}$ which are a solution of the functional equation (where $f^{-1}$ denotes inverse of the function) $$f^{-1}(-x)=-f(x)$$ with the additional condition $$\left. \frac{df}{dx} \right|_{x=0}=1?$$