How do you differentiate f(x)= x/(x-3 ) using the quotient rule?

Mar 2, 2017

$f ' \left(x\right) = - \frac{3}{x - 3} ^ 2$

Explanation:

If $f \left(x\right) = g \frac{x}{h \left(x\right)}$ then the quotient rule states that:

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

In this example: $f \left(x\right) = \frac{x}{x - 3}$

Hence: $g \left(x\right) = x$ and $h \left(x\right) = \left(x - 3\right)$

Applying the quotient in this example:

$f ' \left(x\right) = \frac{\left(x - 3\right) \cdot 1 - x \cdot 1}{x - 3} ^ 2$

$= \frac{x - 3 - x}{x - 3} ^ 2$

$= - \frac{3}{x - 3} ^ 2$