${\left(f\left(x_0\right)-g_{x_0}\right)}^{{}^{\prime }}=\underset{x\to x_0}{\lim }\frac{\left[f\left(x\right)-g\left(x\right)\right]-\left[f\left(x_0\right)-g_{x_0}\right]}{x-x_0}=\underset{x\to x_0}{\lim }\frac{f\left(x\right)-f\left(x_0\right)-\left(g\left(x\right)-g\left(x_0\right)\right)}{x-x_0}=\underset{x\to x_0}{\lim }\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}-\underset{x\to x_0}{\lim }\frac{g\left(x\right)-g\left(x_0\right)}{x-x_0}=f^{{}^{\prime }}\left(x_0\right)-g^{{}^{\prime }}\left(x_0\right)$