مشتقة الدالة f(x)=1/x هي: f’(x) = -1/x^2 لكل x غير صفر

مشتقة الدالة f(x)=1/x

مشتقة الدالة $f(x)=\frac{1}{x}$ هي:

\[\forall x \in \mathbb{R}^* , f'(x) = -\dfrac{1}{x^2}\]

الإثبات

ليكن $x \in \mathbb{R}^*$

\[\begin{aligned} \frac{df}{dx} &= \lim_{h \rightarrow 0} \frac{\dfrac{1}{x+h} - \dfrac{1}{x}}{h}\\ &= \lim_{h \rightarrow 0} \frac{\dfrac{1}{x+h} \cdot \dfrac{x}{x} - \dfrac{1}{x} \cdot \dfrac{x+h}{x+h}}{h}\\ &= \lim_{h \rightarrow 0} \frac{\dfrac{x - (x+h)}{x(x+h)}}{h}\\ &= \lim_{h \rightarrow 0} \frac{\dfrac{-h}{x(x+h)}}{h}\\ &= \lim_{h \rightarrow 0} \frac{\dfrac{-1}{x(x+h)}}{1}\\ &= \lim_{h \rightarrow 0} \frac{-1}{x(x+h)} = \frac{-1}{x(x+0)}\\ &= -\frac{1}{x^{2}} \end{aligned}\]

وبالتالي:

\[\forall x \in \mathbb{R}^* , f'(x) = -\dfrac{1}{x^2}\]