Let $A$ be a $C^*$-algebra, $f \colon \mathbb{R} \to \mathbb{C}$ be a continuous function, and $\tilde{f} \colon A_{\text{sa}} \to A$ be the functional calculus map $A_{\text{sa}} \ni a \mapsto f(a) \in A$. It is elementary to show that $\tilde{f}$ is continuous, so it is natural to wonder how the differentiability properties of $f$ relate/transfer to those of $\tilde{f}$. This turns out to be a delicate, complicated problem. In this talk, I introduce a rich class $NC^k(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of noncommutative $C^k$ functions $f$ such that $\tilde{f}$ is $k$-times differentiable. I shall also discuss the interesting objects, called multiple operator integrals, used to express the derivatives of $\tilde{f}$.