f(x) = 1/x
a = 2
Use the definition:
f' (a) = lim as x approaches a of f(x) - f(a) / x - a
Can someone help?
$\displaystyle f(x) = \frac{1}{x}$
$\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$
$\displaystyle = \lim_{h \to 0}\frac{\frac{1}{x + h} - \frac{1}{x}}{h}$
$\displaystyle = \lim_{h \to 0}\frac{\frac{x - (x + h)}{x(x + h)}}{h}$
$\displaystyle = \lim_{h \to 0}\frac{\frac{-h}{x(x + h)}}{h}$
$\displaystyle = \lim_{h \to 0}\left(\frac{-h}{x(x + h)}\cdot\frac{1}{h}\right)$
$\displaystyle = \lim_{h \to 0}\frac{-1}{x(x + h)}$
$\displaystyle = -\frac{1}{x^2}$.
Therefore
$\displaystyle f'(a) = -\frac{1}{a^2}$.