Question

Find the derivative of

`f(x) = (6)/(x)`
at x = 2.

Answer 1

`f'(x_0)=\lim\limits_(h\to0)(f(x_0+h)-f(h))/(h)=\lim\limits_(x\to x_0)(f(x)-f(x_0))/(x-x_0)\\\\f(x)=(6)/(x);\ x_0=2\\\\subtitute\\\\f'(2)=\lim\limits_(x\to2)((6)/(x)-(6)/(2))/(x-2)=\lim\limits_(x\to2)((6)/(x)-3)/(x-2)=\lim\limits_(x\to2)((6-3x)/(x))/(x-2)=\lim\limits_(x\to 2)(6-3x)/(x(x-2))\\\\=\lim\limits_(x\to2)(-3(x-2))/(x(x-2))=\lim\limits_(x\to2)(-3)/(x)=-(3)/(2)=-1.5\\\\\\An swer:\boxed{f'(2)=-1.5}`