Derivative of $E: \mathbb{R}^3\to \mathbb{R}^3, E(x)=Ax+c$

Question

Derivative of $E: \mathbb{R}^3\to \mathbb{R}^3, E(x)=Ax+c$

In general, how can I derive an affine transformation with $A\in \operatorname{GL}(3,\mathbb{R})$, $x,c\in \mathbb{3}$ with respect to $x$?

Answer 1

For any $x,h$ you have

$$E(x+h) - E(x) = Ah$$ where $h \mapsto Ah$ is linear. Therefore the derivative of $E$ at any point $x$ is the linear map $h \mapsto Ah$.