\(f'(x_0)=\lim\limits_{x\to x_0}\dfrac{f(x)-f(x_0)}{x-x_0}\\\\f(x)=\dfrac{5}{x};\ x_0=-1\\\\subtitute\\\\f'(-1)=\lim\limits_{x\to-1}\dfrac{\frac{5}{x}-\frac{5}{-1}}{x-(-1)}=\lim\limits_{x\to -1}\dfrac{\frac{5}{x}+5}{x+1}=\lim\limits_{x\to-1}\dfrac{\frac{5}{x}+\frac{5x}{x}}{x+1}\\\\=\lim\limits_{x\to-1}\dfrac{\frac{5+5x}{x}}{x+1}=\lim\limits_{x\to-1}\left(\dfrac{5+5x}{x}\cdot\dfrac{1}{x+1}\right)\\\\=\lim\limits_{x\to-1}\dfrac{5(x+1)}{x(x+1)}=\lim\limits_{x\to-1}\dfrac{5}{x}=\dfrac{5}{-1}=-5\)
