[tex]\displaystyle\\\boxed{f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}}\\\\\\\\f(x)=\dfrac{1}{x}\\\\\\f'(x)=\lim_{h\to0}\dfrac{\dfrac{1}{x+h}-\dfrac{1}{x}}{h}\\\\f'(x)=\lim_{h\to0}\dfrac{\dfrac{x}{x(x+h)}-\dfrac{x+h}{x(x+h)}}{h}\\\\f'(x)=\lim_{h\to0}\dfrac{\dfrac{-h}{x(x+h)}}{h}\\\\f'(x)=\lim_{h\to0}-\dfrac{h}{hx(x+h)}\\\\f'(x)=\lim_{h\to0}-\dfrac{1}{x(x+h)}\\\\f'(x)=-\dfrac{1}{x(x+0)}\\\\f'(x)=-\dfrac{1}{x^2}[/tex]
