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Identical expressions
(log(4x- two))/(ctg2x)
( logarithm of (4x minus 2)) divide by (ctg2x)
( logarithm of (4x minus two)) divide by (ctg2x)
log4x-2/ctg2x
(log(4x-2)) divide by (ctg2x)
Similar expressions
(log(4x+2))/(ctg2x)

The derivative of the function/ (log(4x-2))/(ctg2x)

Derivative of (log(4x-2))/(ctg2x)

The solution

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log(4*x - 2)
------------
  cot(2*x)

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(4 x - 2 \right)}$ and $g{\left(x \right)} = \cot{\left(2 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 4 x - 2$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(4 x - 2\right)$ :
  1. Differentiate $4 x - 2$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $4$
    2. The derivative of the constant $-2$ is zero.
    The result is: $4$
  The result of the chain rule is:
  
  $\frac{4}{4 x - 2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(2 x \right)} = \frac{1}{\tan{\left(2 x \right)}}$
  2. Let $u = \tan{\left(2 x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(2 x \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(2 x \right)} = \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \cos{\left(2 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $- 2 \sin{\left(2 x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
    The result of the chain rule is:
    
    $- \frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)} \tan^{2}{\left(2 x \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(2 x \right)} = \frac{\cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(2 x \right)}$ and $g{\left(x \right)} = \sin{\left(2 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 2 x$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $- 2 \sin{\left(2 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 2 x$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{- 2 \sin^{2}{\left(2 x \right)} - 2 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}$
Now plug in to the quotient rule:

$\frac{\frac{\left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \log{\left(4 x - 2 \right)}}{\cos^{2}{\left(2 x \right)} \tan^{2}{\left(2 x \right)}} + \frac{4 \cot{\left(2 x \right)}}{4 x - 2}}{\cot^{2}{\left(2 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                     /         2     \             
        4            \2 + 2*cot (2*x)/*log(4*x - 2)
------------------ + ------------------------------
(4*x - 2)*cot(2*x)                2                
                               cot (2*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of (log(4x-2))/(ctg2x)

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Piecewise:

The solution

Method #1

Method #2