Derivative of y=x/(x-1)*(x-4)

1. Apply the quotient rule, which is:

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

   $f{\left(x \right)} = x \left(x - 4\right)$ and $g{\left(x \right)} = x - 1$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the product rule:

      $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

      $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

      1. Apply the power rule: $x$ goes to $1$

      $g{\left(x \right)} = x - 4$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Differentiate $x - 4$ term by term:

         1. The derivative of the constant $-4$ is zero.

         2. Apply the power rule: $x$ goes to $1$

         The result is: $1$

      The result is: $2 x - 4$

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $x - 1$ term by term:

      1. The derivative of the constant $-1$ is zero.

      2. Apply the power rule: $x$ goes to $1$

      The result is: $1$

   Now plug in to the quotient rule:

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

2. Now simplify:

   $$\frac{x^{2} - 2 x + 4}{x^{2} - 2 x + 1}$$

The answer is: