Is the derivative of a bump function still a bump function? My question is rather simple : are derivatives of bump functions still bump functions ?
For example, for a bump function $u\in D(\mathbb{R}^d)$, that is,
$$u \in C^\infty|\;\text{supp}\;u\subseteq K : \mathbb{R}^d \to \mathbb{R}$$
Is this always true for all derivatives?
$$\frac{\partial u}{\partial x_j} \in D(\mathbb{R}^d)$$
I'm sure it is and couldn't imagine the contrary but I have an unexplainable doubt.

Yes, because, obviously, $\operatorname{supp}\partial u/\partial x_j\subset\operatorname{supp}u$.
