2.2 Lösning 7

\[ \Delta y = f(a + h) \, - \, fa) = (a + h)^2 - a^2 = a^2 + 2\,a\,h + h^2 - a^2 = 2\,a\,h + h^2 \]

\[ \Delta x \, = \, a + h \, - \, a \, = \, h \]

\[ {\Delta y \over \Delta x} \, = \, {f(x_1 + h) \, - \, f(x_1) \over h} \]

Om vi tillämpar derivatans definition på \( f(x) = x^2\, \) kan vi skriva\[ f\,'(x) = \lim_{h \to 0} {f(x+h) - f(x) \over h} = \lim_{h \to 0} {(x+h)^2 - x^2 \over h} = \lim_{h \to 0} {(x^2 + 2\,x\,h + h^2) - x^2 \over h} = \lim_{h \to 0} {2\,x\,h + h^2 \over h} = \]

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