The definition of derivative is
#f'(x) = \lim_{x\to0}\frac{f(x+h)-f(x)}{h}#
so, if #f(x) = \frac{5-x}{x}#, the definition leads to
#f'(x) = \lim_{x\to0}\frac{\frac{5-x-h}{x+h}-\frac{5-x}{x}}{h}#
Manipulate the numerator to get
#\frac{5-x-h}{x+h}-\frac{5-x}{x} = \frac{(5-x-h)x - (5-x)(x+h)}{x(x+h)#
#=\frac{5x-x^2-hx - 5x+x^2-5h+hx}{x(x+h)} = \frac{-5h}{x(x+h)}#
so,
#f'(x) = \lim_{x\to0}\frac{\frac{5-x-h}{x+h}-\frac{5-x}{x}}{h} = \frac{-5h}{hx(x+h)} = \frac{-5}{x(x+h)}#
When #h \to 0#, #x+h \to x#, so
#f'(x) = \lim_{x\to0}\frac{-5}{x(x+h)} = \frac{-5}{x^2}#
