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Identical expressions
ln(sin(2x))/(one /tan(2x))
ln( sinus of (2x)) divide by (1 divide by tangent of (2x))
ln( sinus of (2x)) divide by (one divide by tangent of (2x))
lnsin2x/1/tan2x
ln(sin(2x)) divide by (1 divide by tan(2x))

The derivative of the function/ ln(sin(2x))/(1/tan(2x))

Derivative of ln(sin(2x))/(1/tan(2x))

The solution

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log(sin(2*x))
-------------
  /   1    \ 
  |--------| 
  \tan(2*x)/

$$\frac{\log{\left(\sin{\left(2 x \right)} \right)}}{\frac{1}{\tan{\left(2 x \right)}}}$$

log(sin(2*x))/1/tan(2*x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(\sin{\left(2 x \right)} \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(2 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(2 x \right)}$ :
  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 $\frac{d}{d x} 2 x$ :
    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: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 x \right)}$
  The result of the chain rule is:
  
  $\frac{2 \cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
$g{\left(x \right)} = \frac{1}{\frac{1}{\tan{\left(2 x \right)}}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{1}{\tan{\left(2 x \right)}}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{\tan{\left(2 x \right)}}$ :
  1. Let $u = \tan{\left(2 x \right)}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. 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 $\frac{d}{d x} f{\left(x \right)}$ :
      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:
        
        $2 \cos{\left(2 x \right)}$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      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:
        
        $- 2 \sin{\left(2 x \right)}$
      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:
    
    $- \frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)} \tan^{2}{\left(2 x \right)}}$
  The result of the chain rule is:
  
  $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
The result is: $\frac{\left(2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \log{\left(\sin{\left(2 x \right)} \right)}}{\cos^{2}{\left(2 x \right)}} + \frac{2 \cos{\left(2 x \right)} \tan{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /          2     \                 2*cos(2*x)*tan(2*x)
- \-2 - 2*tan (2*x)/*log(sin(2*x)) + -------------------
                                           sin(2*x)

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

Simplify

The second derivative [src]

  /  /       2     \              /       2     \                                                    \
  |  |    cos (2*x)|            2*\1 + tan (2*x)/*cos(2*x)     /       2     \                       |
4*|- |1 + ---------|*tan(2*x) + -------------------------- + 2*\1 + tan (2*x)/*log(sin(2*x))*tan(2*x)|
  |  |       2     |                     sin(2*x)                                                    |
  \  \    sin (2*x)/                                                                                 /

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

Simplify

The third derivative [src]

  /                                                                                            /       2     \                                                        \
  |                                                                                            |    cos (2*x)|                                                        |
  |                                                                                          2*|1 + ---------|*cos(2*x)*tan(2*x)                                      |
  |                    /       2     \                                                         |       2     |                       /       2     \                  |
  |    /       2     \ |    cos (2*x)|     /       2     \ /         2     \                   \    sin (2*x)/                     6*\1 + tan (2*x)/*cos(2*x)*tan(2*x)|
8*|- 3*\1 + tan (2*x)/*|1 + ---------| + 2*\1 + tan (2*x)/*\1 + 3*tan (2*x)/*log(sin(2*x)) + ----------------------------------- + -----------------------------------|
  |                    |       2     |                                                                     sin(2*x)                              sin(2*x)             |
  \                    \    sin (2*x)/                                                                                                                                /

$$8 \left(- 3 \left(1 + \frac{\cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) + \frac{2 \left(1 + \frac{\cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}\right) \cos{\left(2 x \right)} \tan{\left(2 x \right)}}{\sin{\left(2 x \right)}} + 2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(3 \tan^{2}{\left(2 x \right)} + 1\right) \log{\left(\sin{\left(2 x \right)} \right)} + \frac{6 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cos{\left(2 x \right)} \tan{\left(2 x \right)}}{\sin{\left(2 x \right)}}\right)$$

Simplify

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Derivative of ln(sin(2x))/(1/tan(2x))

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The solution