Linear response for intermittent maps with critical point

We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables $\phi: [0,1] \to \mathbb{R}$ the bivariate map $(\alpha, \beta) \mapsto \int_0^1 \phi \, d\mu_{\alpha,\beta}$, where $\mu_{\alpha, \beta}$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.