Derivative of ln(ctg2x)

The solution

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log(cot(2*x))

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

log(cot(2*x))

Detail solution

Let $u = \cot{\left(2 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cot{\left(2 x \right)}$ :
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:
    
    $\frac{- 2 \sin^{2}{\left(2 x \right)} - 2 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          2     
-2 - 2*cot (2*x)
----------------
    cot(2*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(ctg2x)

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Piecewise:

The solution

Method #1

Method #2