Question

Plz any help on how to find derivatives would be fantastic.

Find the derivative of f(x)=5/x at x=-1

Answer 1

`f'(x_0)=\lim\limits_(x\to x_0)(f(x)-f(x_0))/(x-x_0)\\\\f(x)=(5)/(x);\ x_0=-1\\\\subtitute\\\\f'(-1)=\lim\limits_(x\to-1)((5)/(x)-(5)/(-1))/(x-(-1))=\lim\limits_(x\to -1)((5)/(x)+5)/(x+1)=\lim\limits_(x\to-1)((5)/(x)+(5x)/(x))/(x+1)\\\\=\lim\limits_(x\to-1)((5+5x)/(x))/(x+1)=\lim\limits_(x\to-1)\left((5+5x)/(x)\cdot(1)/(x+1)\right)\\\\=\lim\limits_(x\to-1)(5(x+1))/(x(x+1))=\lim\limits_(x\to-1)(5)/(x)=(5)/(-1)=-5`