Difference quotient for $f{\left(x \right)} = \frac{2}{x}$

Find the difference quotient for $f{\left(x \right)} = \frac{2}{x}$.

The difference quotient is given by $\frac{f{\left(x + h \right)} - f{\left(x \right)}}{h}$.

To find $f{\left(x + h \right)}$, plug $x + h$ instead of $x$: $f{\left(x + h \right)} = \frac{2}{x + h}$.

Finally, $\frac{f{\left(x + h \right)} - f{\left(x \right)}}{h} = \frac{\frac{2}{x + h} - \frac{2}{x}}{h} = - \frac{2}{x \left(h + x\right)}$.

The difference quotient for $f{\left(x \right)} = \frac{2}{x}$A is $- \frac{2}{x \left(h + x\right)}$A.