\[\begin{aligned} &\begin{cases} f(y_n)-f(x_0)=f'(x_0)(y_n-x_0)+\alpha_n(y_n-x_0) \\ f(x_n)-f(x_0)=f'(x_0)(x_n-x_0)+\beta_n(x_n-x_0) \\ \end{cases} \\ \Rightarrow & \frac{f(y_n)-f(x_n)}{y_n-x_n}=f'(x_0)+\frac{\alpha_n(y_n-x_0)}{y_n-x_n}+\frac{\beta_n(x_n-x_0)}{y_n-x_N} \\ \Rightarrow & f'(x_0)=\lim_{n \to \infty} \frac{f(y_n)-f(x_n)}{y_n-x_n} \end{aligned} \]