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Graphing y =:
2*ln(x/(x-4))-3
Identical expressions
two *ln(x/(x- four))- three
2 multiply by ln(x divide by (x minus 4)) minus 3
two multiply by ln(x divide by (x minus four)) minus three
2ln(x/(x-4))-3
2lnx/x-4-3
2*ln(x divide by (x-4))-3
Similar expressions
2*ln(x/(x+4))-3
2*ln(x/(x-4))+3

The derivative of the function/ 2*ln(x/(x-4))-3

Derivative of 2*ln(x/(x-4))-3

The solution

You have entered [src]

     /  x  \    
2*log|-----| - 3
     \x - 4/

$$2 \log{\left(\frac{x}{x - 4} \right)} - 3$$

2*log(x/(x - 4)) - 3

Detail solution

Differentiate $2 \log{\left(\frac{x}{x - 4} \right)} - 3$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{x}{x - 4}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{x - 4}$ :
    1. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = x$ and $g{\left(x \right)} = x - 4$ .
      
      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 $x - 4$ term by term:
        
        The derivative of the constant $-4$ is zero.
        
        Apply the power rule: $x$ goes to $1$
        
        The result is: $1$
      Now plug in to the quotient rule:
      
      $- \frac{4}{\left(x - 4\right)^{2}}$
    The result of the chain rule is:
    
    $- \frac{4 \left(x - 4\right)}{x \left(x - 4\right)^{2}}$
  So, the result is: $- \frac{8 \left(x - 4\right)}{x \left(x - 4\right)^{2}}$
2. The derivative of the constant $-3$ is zero.
The result is: $- \frac{8 \left(x - 4\right)}{x \left(x - 4\right)^{2}}$
Now simplify:

$- \frac{8}{x \left(x - 4\right)}$

The answer is:

$- \frac{8}{x \left(x - 4\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of 2*ln(x/(x-4))-3

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The solution