Derivative of a function $f: \mathbb{R} \rightarrow B$, $B$ is a $C^*$ algebra

Question

Let $B$ be a $C^*$ algebra. Let $h \in B$ be a self adjoint element. Then by the continuous functional calculus we can talk about the element $e^{ith}$ where $i \in \mathbb{C}, t \in \mathbb{R}$. Then what is the concept of derivative of a $C^*$ valued function from real numbers? Is the derivative of $e^{ith}$ at $t=0$ is ih? Is chain rule for derivative of product is also valid?