When does separate Frechet differentability imply joint Frechet differentiability?

Question

Let $f\colon X \times Y \to Z$ be a mapping between Banach spaces. If I know that $f(\cdot,y)$ and $f(x,\cdot)$ are Frechet differentiable or $C^1$ functions (for fixed $x$ and $y$), what other conditions do I need to ensure that $f$ is Frechet or $C^1$ on the product space? Does continuity of the partial derivatives suffice?

Looking for a reference request or proof.

Answer 1

Answer can be found in the final theorem of these notes.