2.3a Lösning 10

\[ f(x+h) - f(x) = {1 \over x+h} - {1 \over x} = {x \over x\,(x+h)} - {x+h \over x\,(x+h)} = {x - (x+h) \over x\,(x+h)} = {x - x - h \over x\,(x+h)} = {- h \over x\,(x+h)} \]

\[ f\,'(x) = \lim_{h \to 0} {f(x+h) - f(x) \over h} = \lim_{h \to 0} {{- h \over x\,(x+h)} \over h} = \lim_{h \to 0} {{- h \over x\,(x+h)} \over {h \over 1}} = \lim_{h \to 0} \; {- h\cdot 1 \over h\cdot x\,(x+h)} = \]

\[ = \lim_{h \to 0} \; {- 1 \over x\,(x+h)} = \lim_{h \to 0} \; {- 1 \over x^2 + x\,h} = - \, {1 \over x^2} \]