Let $F \in \operatorname{E}(n)$ be a Euclidean motion, i.e. $F = Ax + b$, where $A \in \operatorname{O}(n)$ and $b \in \mathbb{R}^n$ and let $c:I \subset \mathbb{R} \to \mathbb{R}^n$ be a curve parametrized by arc length. I want to calculate the derivative of $F \circ c$. Here's what I get by using the chain rule and the fact that $F' = A$:
$\frac{d}{dt} F(c(t)) = \frac{d}{dt} \left( Ac(t) + b \right) = \frac{d}{dt} Ac(t) + \frac{d}{dt}b = c'(t)Ac(t)$.
However, my text book says that $(F \circ c)'(t) = Ac'(t)$. Where's my error and what is the correct way to derive this function? Any help is appreciated.