# Math Help - Finding the derivative

1. ## Finding the derivative

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?

2. Originally Posted by Morgan82
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?
$f(x) = \frac{1}{x}$

$f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

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

$= \lim_{h \to 0}\frac{\frac{x - (x + h)}{x(x + h)}}{h}$

$= \lim_{h \to 0}\frac{\frac{-h}{x(x + h)}}{h}$

$= \lim_{h \to 0}\left(\frac{-h}{x(x + h)}\cdot\frac{1}{h}\right)$

$= \lim_{h \to 0}\frac{-1}{x(x + h)}$

$= -\frac{1}{x^2}$.

Therefore

$f'(a) = -\frac{1}{a^2}$.