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Identical expressions
x+ five +6ln((x)/(x- one))
x plus 5 plus 6ln((x) divide by (x minus 1))
x plus five plus 6ln((x) divide by (x minus one))
x+5+6lnx/x-1
x+5+6ln((x) divide by (x-1))
Similar expressions
x+5-6ln((x)/(x-1))
x+5+6ln((x)/(x+1))
x-5+6ln((x)/(x-1))

The derivative of the function/ x+5+6ln((x)/(x-1))

Derivative of x+5+6ln((x)/(x-1))

The solution

You have entered [src]

             /  x  \
x + 5 + 6*log|-----|
             \x - 1/

$$\left(x + 5\right) + 6 \log{\left(\frac{x}{x - 1} \right)}$$

x + 5 + 6*log(x/(x - 1))

Detail solution

Differentiate $\left(x + 5\right) + 6 \log{\left(\frac{x}{x - 1} \right)}$ term by term:
1. Differentiate $x + 5$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $5$ is zero.
  The result is: $1$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{x}{x - 1}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{x - 1}$ :
    1. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = x$ and $g{\left(x \right)} = x - 1$ .
      
      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. Differentiate $x - 1$ term by term:
        
        The derivative of the constant $-1$ is zero.
        
        Apply the power rule: $x$ goes to $1$
        
        The result is: $1$
      Now plug in to the quotient rule:
      
      $- \frac{1}{\left(x - 1\right)^{2}}$
    The result of the chain rule is:
    
    $- \frac{x - 1}{x \left(x - 1\right)^{2}}$
  So, the result is: $- \frac{6 \left(x - 1\right)}{x \left(x - 1\right)^{2}}$
The result is: $1 - \frac{6 \left(x - 1\right)}{x \left(x - 1\right)^{2}}$
Now simplify:

$\frac{x \left(x - 1\right) - 6}{x \left(x - 1\right)}$

The answer is:

$\frac{x \left(x - 1\right) - 6}{x \left(x - 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /       x   \ /1     1   \
6*|-1 + ------|*|- + ------|
  \     -1 + x/ \x   -1 + x/
----------------------------
             x

$$\frac{6 \left(\frac{x}{x - 1} - 1\right) \left(\frac{1}{x - 1} + \frac{1}{x}\right)}{x}$$

Simplify

The third derivative [src]

   /       x   \ /  1        1           1     \
12*|-1 + ------|*|- -- - --------- - ----------|
   \     -1 + x/ |   2           2   x*(-1 + x)|
                 \  x    (-1 + x)              /
------------------------------------------------
                       x

$$\frac{12 \left(\frac{x}{x - 1} - 1\right) \left(- \frac{1}{\left(x - 1\right)^{2}} - \frac{1}{x \left(x - 1\right)} - \frac{1}{x^{2}}\right)}{x}$$

Simplify

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Derivative of x+5+6ln((x)/(x-1))

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The solution