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Identical expressions
(log(three)(4x- two))/ctg(2x)
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Similar expressions
(log(3)(4x+2))/ctg(2x)

The derivative of the function/ (log(3)(4x-2))/ctg(2x)

Derivative of (log(3)(4x-2))/ctg(2x)

The solution

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

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \left(4 x - 2\right) \log{\left(3 \right)}$ and $g{\left(x \right)} = \cot{\left(2 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Differentiate $4 x - 2$ term by term:
    1. The derivative of the constant $-2$ is zero.
    2. 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: $4$
    The result is: $4$
  So, the result is: $4 \log{\left(3 \right)}$
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:

$\frac{\frac{\left(4 x - 2\right) \left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \log{\left(3 \right)}}{\cos^{2}{\left(2 x \right)} \tan^{2}{\left(2 x \right)}} + 4 \log{\left(3 \right)} \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)}}$
Now simplify:

$\frac{4 \left(4 x + \sin{\left(4 x \right)} - 2\right) \log{\left(3 \right)}}{\cos{\left(4 x \right)} + 1}$

The answer is:

$\frac{4 \left(4 x + \sin{\left(4 x \right)} - 2\right) \log{\left(3 \right)}}{\cos{\left(4 x \right)} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           /         2     \                 
4*log(3)   \2 + 2*cot (2*x)/*(4*x - 2)*log(3)
-------- + ----------------------------------
cot(2*x)                  2                  
                       cot (2*x)

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

Simplify

The second derivative [src]

                   /                      /            2     \\       
   /       2     \ |   1                  |     1 + cot (2*x)||       
16*\1 + cot (2*x)/*|-------- + (-1 + 2*x)*|-1 + -------------||*log(3)
                   |cot(2*x)              |          2       ||       
                   \                      \       cot (2*x)  //       
----------------------------------------------------------------------
                               cot(2*x)

$$\frac{16 \left(\left(2 x - 1\right) \left(\frac{\cot^{2}{\left(2 x \right)} + 1}{\cot^{2}{\left(2 x \right)}} - 1\right) + \frac{1}{\cot{\left(2 x \right)}}\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) \log{\left(3 \right)}}{\cot{\left(2 x \right)}}$$

Simplify

The third derivative [src]

   /                                                                                           /            2     \\       
   |                                                                           /       2     \ |     1 + cot (2*x)||       
   |           /                                   2                    3\   3*\1 + cot (2*x)/*|-1 + -------------||       
   |           |                    /       2     \      /       2     \ |                     |          2       ||       
   |           |         2        5*\1 + cot (2*x)/    3*\1 + cot (2*x)/ |                     \       cot (2*x)  /|       
32*|(-1 + 2*x)*|2 + 2*cot (2*x) - ------------------ + ------------------| + --------------------------------------|*log(3)
   |           |                         2                    4          |                  cot(2*x)               |       
   \           \                      cot (2*x)            cot (2*x)     /                                         /

$$32 \left(\left(2 x - 1\right) \left(\frac{3 \left(\cot^{2}{\left(2 x \right)} + 1\right)^{3}}{\cot^{4}{\left(2 x \right)}} - \frac{5 \left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}{\cot^{2}{\left(2 x \right)}} + 2 \cot^{2}{\left(2 x \right)} + 2\right) + \frac{3 \left(\frac{\cot^{2}{\left(2 x \right)} + 1}{\cot^{2}{\left(2 x \right)}} - 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)}{\cot{\left(2 x \right)}}\right) \log{\left(3 \right)}$$

Simplify

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Derivative of (log(3)(4x-2))/ctg(2x)

The graph:

Piecewise:

The solution

Method #1

Method #2