It is my understanding that and endomorphism is a morphism from a mathematical object to itself. For instance in a vector space $V$ if $f$ is an endomorphism, $f: V \rightarrow V$.
However the derivative of a function in a space $V$ maps to a subset of $V$. So, with this in mind, is the derivative an endomorphism?
An endomorphism need not to be surjective. So whenever you have a morphism $g$ mapping an object $X$ to a sub-object $Y\subseteq X$, you can regard $g$ as an endomorphism $g\colon X\to X$.
It needn't be. It depends what the domain is. For instance, if the domain consists of the space of differentiable functions on the real numbers, there are examples of such functions whose derivatives are not differentiable. A standard example is $f(x)=\begin{cases}x^2\sin1/x,\,x\ne0\\0,\,x=0\end{cases}$.
