I have a question about the following. A representation $D$ of a matrix Lie group $G$ is a homomorphism from $G$ to $GL(n,K)$ which is also differentiable function of the parameters $\theta_1,...,\theta_k$. The parameters are real parameters and $g = g(\theta_1,...,\theta_k)$ for $g \in G$ and the parameters depend on the element $g$ you look at. This is the definition I read in the lecture notes I use to study (matrix) Lie groups.
Question: How is differentiability of a representation well-defined when we talk about a map $G \rightarrow GL(n,K)$ that we can write as $\mathcal{R}^k \rightarrow GL(n,K)$ by identifying $D(g(\theta_1,...,\theta_k)) = f(\theta_1,...,\theta_k)$ ?
Note: In the lecture notes they just took the derivative of $f$ as follows for the case of $U(1)$ so use $\phi$ as parameter:
$$f^\prime(0)= \lim_{\phi \rightarrow 0} \frac{ f(\phi) - f(0)}{\phi}$$.
So how is taking the derivative of a representation well-defined in this way?