How do you find the derivative of f(x)=1/x using the limit definition?

Jul 9, 2016

$f ' \left(x\right) = - \frac{1}{x} ^ 2$

Explanation:

$f ' \left(x\right) = {\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

$= {\lim}_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h}$

Resolve the numerator into one fraction:

$= {\lim}_{h \to 0} \frac{\frac{x}{x \left(x + h\right)} - \frac{x + h}{x \left(x + h\right)}}{h} = \frac{\frac{- h}{x \left(x + h\right)}}{h}$

$= {\lim}_{h \to 0} \frac{- 1}{x \left(x + h\right)} = - \frac{1}{x} ^ 2$