  Let $\gamma_1,\gamma_2:[0,T]\to \overline{\mathbb{D}}\setminus\{0\}$ be
parametrizations of two slits $\Gamma_1:=\gamma(0,T], \Gamma_2=\gamma_2(0,T]$
such that $\Gamma_1$ and $\Gamma_2$ are disjoint. \\ Let $g_t$ to be the unique
normalized conformal mapping from $\mathbb{D}\setminus (\gamma_1[0,t]\cup
\gamma_2[0,t])$ onto $\mathbb{D}$ with $g_t(0)=0,$ $g'_t(0)>0$. Furthermore,
for $k=1,2$, denote by $h_{k;t}$ the unique normalized conformal mapping from
$\mathbb{D}\setminus \gamma_k[0,t]$ onto $\mathbb{D}$ with $h_{k;t}(0)=0,$
${h'_{k;t}(0)}>0$.\\ Loewner's famous theorem (\cite{Loewner:1923}) can be
stated in the following way: The function $t\mapsto h_{k;t}$ is differentiable
at $t_0$ if and only if $t\mapsto \log(h_{k;t}'(0))$ is differentiable at
$t_0$.\\ In this paper we compare the differentiability of $t\mapsto h_{k;t}$
with that of $t\mapsto g_t.$ We show that the situation is more complicated in
the case $t_0=0$ with $\gamma_1(0)=\gamma_2(0).$\\ Furthermore, we also look at
this problem in the case of a multiply connected domain with its corresponding
Komatu-Loewner equation.
