Question

Find the derivative of f(x) = 8/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)=(8)/(x);\ x_0=-1\\\\subtitute\\\\f'(-1)=\lim\limits_(x\to-1)((8)/(x)-(8)/(-1))/(x-(-1))=\lim\limits_(x\to -1)((8)/(x)+8)/(x+1)=\lim\limits_(x\to-1)((8)/(x)+(8x)/(x))/(x+1)\\\\=\lim\limits_(x\to-1)((8+8x)/(x))/(x+1)=\lim\limits_(x\to-1)\left((8+8x)/(x)\cdot(1)/(x+1)\right)\\\\=\lim\limits_(x\to-1)(8(x+1))/(x(x+1))=\lim\limits_(x\to-1)(8)/(x)=(8)/(-1)=-8`