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Graphing y =:
x/(x-1)/(x-4)
Identical expressions
y=x/(x- one)/(x- four)
y equally x divide by (x minus 1) divide by (x minus 4)
y equally x divide by (x minus one) divide by (x minus four)
y=x/x-1/x-4
y=x divide by (x-1) divide by (x-4)
Similar expressions
y=x/(x+1)/(x-4)
y=x/(x-1)/(x+4)

The derivative of the function/ y=x/(x-1)/(x-4)

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

The solution

You have entered [src]

/  x  \
|-----|
\x - 1/
-------
 x - 4

$$\frac{x \frac{1}{x - 1}}{x - 4}$$

(x/(x - 1))/(x - 4)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x - 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  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: $1$
  $g{\left(x \right)} = x - 4$ ; 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: $2 x - 5$
Now plug in to the quotient rule:

$\frac{- x \left(2 x - 5\right) + \left(x - 4\right) \left(x - 1\right)}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}$

The answer is:

$\frac{- x \left(2 x - 5\right) + \left(x - 4\right) \left(x - 1\right)}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1        x                       
----- - --------                   
x - 1          2                   
        (x - 1)           x        
---------------- - ----------------
     x - 4                        2
                   (x - 1)*(x - 4)

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

Simplify

The second derivative [src]

  /                   x             x   \
  |            -1 + ------   -1 + ------|
  |    x            -1 + x        -1 + x|
2*|--------- + ----------- + -----------|
  |        2      -1 + x        -4 + x  |
  \(-4 + x)                             /
-----------------------------------------
            (-1 + x)*(-4 + x)

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

Simplify

The third derivative [src]

   /                   x             x                x      \
   |            -1 + ------   -1 + ------      -1 + ------   |
   |    x            -1 + x        -1 + x           -1 + x   |
-6*|--------- + ----------- + ----------- + -----------------|
   |        3            2             2    (-1 + x)*(-4 + x)|
   \(-4 + x)     (-1 + x)      (-4 + x)                      /
--------------------------------------------------------------
                      (-1 + x)*(-4 + x)

$$- \frac{6 \left(\frac{x}{\left(x - 4\right)^{3}} + \frac{\frac{x}{x - 1} - 1}{\left(x - 1\right)^{2}} + \frac{\frac{x}{x - 1} - 1}{\left(x - 4\right) \left(x - 1\right)} + \frac{\frac{x}{x - 1} - 1}{\left(x - 4\right)^{2}}\right)}{\left(x - 4\right) \left(x - 1\right)}$$

Simplify

The graph

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

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The solution