$$\sigma_N (g)'(x_0)=-\int_\pi^\pi F_N'(x_0-t)g(t)dt=\int_{-\pi}^{\pi}F_N'(t)g(x_0-t)dt,$$ where $F_N$ is the Fejér kernel. Also we have $\int_{-\pi}^\pi F_N'(t)dt=0$.
To clarify notations, $\sigma_N(f)(x)=\frac{S_0(f)(x)+\cdots+S_{N-1}(f)(x)}{N}$, $S_n(f)=f*D_n$, $\sigma_N(f)(x)=(f*F_N)(x)$, $D_N(x)=\sum_{n=-N}^N e^{inx}$, and $F_N(x)=\frac{D_0(x)+\cdots+D_{N-1}(x)}{N}$.
Also, $g$ here is any continuous function that is differentiable at $x_0$.
The first paragraph is from a proof in Stein and Shakarchi's Fourier Analysis. But I don't understand why we have the identity and why we have $\int_{-\pi}^\pi F_N'(t)dt=0$. I would greatly appreciate any help.