# How do you find the derivative of f(x)= -x/(x-1)?

Nov 21, 2017

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

#### Explanation:

The most common way to evaluate this derivative is to use the quotient rule, but if we want to avoid using it to differentiate, we can be clever and use a bit of algebraic long division:
$- \frac{x}{x - 1} = - \left(1 + \frac{1}{x - 1}\right) = - \left(1 + {\left(x - 1\right)}^{-} 1\right)$

We can then differentiate this expression using only the chain rule and letting $u = x - 1$:
$- \frac{d}{\mathrm{dx}} \left(1 + {\left(x - 1\right)}^{-} 1\right) = - \frac{d}{\mathrm{du}} \left(1 + {u}^{-} 1\right) \frac{d}{\mathrm{dx}} \left(x - 1\right)$

$= {u}^{-} 2 \cdot x$ (using the power rule)

$= x {\left(x - 1\right)}^{-} 2 = \frac{x}{x - 1} ^ 2$ (resubstitute and simplify)