Let $f$ be a function of a real variable expandable in power series on $\mathbb R$: there exists a sequence $(a_n)_{n\in\mathbb N}$ of reals such that for all $x\in\mathbb R$, one has $$f(x)=\sum_{n\ge0}a_nx^n.$$

Let $(P_n)_{n\in\mathbb N}$ be a sequence of polynomials that converges uniformly towards $f$ on every compact of $\mathbb R$. One assumes that for all $n\in\mathbb N$, one has $P'_n(0)=0$. Is $a_1=0$ ?

If we would work in $\mathbb C$ instead of $\mathbb R$, it would be obvious, but in this setting, I do not know if it is true or not.

Thanks in advance for any hint.