# How do you differentiate f(x)=x/(x-1)^2+x^2-4/(1-2x) using the sum rule?

Apr 14, 2016

$f ' \left(x\right) = - \frac{x + 1}{x - 1} ^ 3 + 2 x - \frac{8}{1 - 2 x} ^ 2$

#### Explanation:

$f \left(x\right) = \frac{x}{x - 1} ^ 2 + {x}^{2} - 4 {\left(1 - 2 x\right)}^{-} 1$

Use Quotients Rule: $f ' \left(x\right) = \frac{g f ' - f g '}{g} ^ 2$ to find the derivative of the first part $\frac{x}{x - 1} ^ 2$

$f ' \left(x\right) = \frac{{\left(x - 1\right)}^{2} - x \left(2 \left(x - 1\right)\right)}{x - 1} ^ 4 + 2 x + \frac{4}{1 - 2 x} ^ 2 \times - 2$

$f ' \left(x\right) = \frac{\left(x - 1\right) \left[x - 1 - 2 x\right]}{x - 1} ^ 4 + 2 x + \frac{4}{1 - 2 x} ^ 2 \times - 2$

$f ' \left(x\right) = - \frac{x + 1}{x - 1} ^ 3 + 2 x - \frac{8}{1 - 2 x} ^ 2$