Verify taking the derivative of this polynomial I've been splitting my head over this, and most likely it's obvious but I can't see it.
Taken from Understanding Analysis, 2nd Edition, Stephen Abbott.

Exercise $\mathbf{6.6.9}$ (Cauchy's Remainder Theorem)
Let $f$ be differentiable $N+1$ times on $(-R,R)$. For each $a\in(-R,R)$, let $S_N(x,a)$ be the partial sum of the Taylor series for $f$ centered at $a;$ in other words, define
$$S_N(x,a)=\sum_{n=0}^{N}{c_n(x-a)^n} \text{ where } c_n=\frac{f^{(n)}(a)}{n!}.$$
Let $E_N(x,a)=f(x)-S_N(x,a).$ Now fix $x\neq0$ in $(-R,R)$ and consider $E_N(x,a)$ as a function of $a$.
Explain why $E_N(x,a)$ is differentiable with respect to $a$, and
show $$E'_N(x,a)=-\frac{f^{(N+1)}(a)}{N!}(x-a)^N. $$

My attempt. $E_N$ is differentiable w.r.t $a$ because $f(x)$ is constant, and $S_N$ is a polynomial in $a$ and hence differentiable. In fact we can say $E_N$ is infinitely differentiable. Now,
\begin{align}
E'_N=-S'_N &=-\sum_{n=1}^{N}{nc_n(x-a)^{n-1}(-1)}\\
&=\sum_{n=1}^{N}{nc_n(x-a)^{n-1}}\\
&=\sum_{n=1}^{N}{n\frac{f^{(n)}(a)}{n!}(x-a)^{n-1}}\\
&=\sum_{n=1}^{N}{\frac{f^{(n)}(a)}{(n-1)!}(x-a)^{n-1}}\\
\end{align}
I'm stuck here. How to proceed?

When you differentiate $S_N$ term-wise, you can't ignore that $c_n$ depends on $a$ (and so $S_N(x, a)$ isn't polynomial in $a$).
Correct way will be
$$E_N' =
\frac{\partial}{\partial a} \left(- \sum_{n = 0}^N\frac{f^{(n)}(a)}{n!}(x - a)^n\right) \\
= - f^{(1)}(a) - \sum_{n=1}^N \frac{\partial}{\partial a} \left(\frac{f^{(n)}(a)}{n!}(x - a)^n\right)\\
= -f^{(1)}(a) - \sum_{n=1}^N \frac{f^{(n + 1)}(a)}{n!}(x - a)^n - \sum_{n=1}^N-\frac{f^{(n)}(a)}{(n - 1)!}(x - a)^{n - 1} \\
= -f^{(1)}(a) - \sum_{n=1}^N \frac{f^{(n + 1)}(a)}{n!}(x - a)^n + \sum_{n = 0}^{N - 1}\frac{f^{(n + 1)}(a)}{n!}(x - a)^n\\
= \color{red}{-f^{(1)}(a)} - \color{blue}{\sum_{n=1}^{N - 1} \frac{f^{(n + 1)}(a)}{n!}(x - a)^n} - \frac{f^{(N + 1)}(a)}{N!}(x - a)^N + \color{red}{f^{(1)}(a)} + \color{blue}{\sum_{n = 1}^{N - 1}\frac{f^{(n + 1)}(a)}{n!}(x - a)^n}\\
= -\frac{f^{(N + 1)}(a)}{N!}(x - a)^N
$$
