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

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

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

The solution

You have entered [src]

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

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

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*(5 - 2*x)   
--------------- + -----------------
(x - 1)*(x - 4)          2        2
                  (x - 1) *(x - 4)

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

Simplify

The second derivative [src]

             /     -5 + 2*x   -5 + 2*x              /  1        1   \\
10 - 4*x + x*|-2 + -------- + -------- + (-5 + 2*x)*|------ + ------||
             \      -1 + x     -4 + x               \-1 + x   -4 + x//
----------------------------------------------------------------------
                                 2         2                          
                         (-1 + x) *(-4 + x)

$$\frac{x \left(\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right) - 2 + \frac{2 x - 5}{x - 1} + \frac{2 x - 5}{x - 4}\right) - 4 x + 10}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}$$

Simplify

The third derivative [src]

       /                                                                                                                        /  1        1   \              /  1        1   \                    \                                                               
       |                                                                                                             (-5 + 2*x)*|------ + ------|   (-5 + 2*x)*|------ + ------|                    |                                                               
       |    8        8                   /    1           1               1        \   3*(-5 + 2*x)   3*(-5 + 2*x)              \-1 + x   -4 + x/              \-1 + x   -4 + x/      4*(-5 + 2*x)  |   3*(-5 + 2*x)   3*(-5 + 2*x)                /  1        1   \
-6 - x*|- ------ - ------ + 2*(-5 + 2*x)*|--------- + --------- + -----------------| + ------------ + ------------ + ---------------------------- + ---------------------------- + -----------------| + ------------ + ------------ + 3*(-5 + 2*x)*|------ + ------|
       |  -1 + x   -4 + x                |        2           2   (-1 + x)*(-4 + x)|            2              2                -1 + x                         -4 + x              (-1 + x)*(-4 + x)|      -1 + x         -4 + x                   \-1 + x   -4 + x/
       \                                 \(-1 + x)    (-4 + x)                     /    (-1 + x)       (-4 + x)                                                                                     /                                                               
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                2         2                                                                                                                         
                                                                                                                        (-1 + x) *(-4 + x)

$$\frac{- x \left(2 \left(2 x - 5\right) \left(\frac{1}{\left(x - 1\right)^{2}} + \frac{1}{\left(x - 4\right) \left(x - 1\right)} + \frac{1}{\left(x - 4\right)^{2}}\right) + \frac{\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right)}{x - 1} - \frac{8}{x - 1} + \frac{3 \left(2 x - 5\right)}{\left(x - 1\right)^{2}} + \frac{\left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right)}{x - 4} - \frac{8}{x - 4} + \frac{4 \left(2 x - 5\right)}{\left(x - 4\right) \left(x - 1\right)} + \frac{3 \left(2 x - 5\right)}{\left(x - 4\right)^{2}}\right) + 3 \left(2 x - 5\right) \left(\frac{1}{x - 1} + \frac{1}{x - 4}\right) - 6 + \frac{3 \left(2 x - 5\right)}{x - 1} + \frac{3 \left(2 x - 5\right)}{x - 4}}{\left(x - 4\right)^{2} \left(x - 1\right)^{2}}$$

Simplify

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

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