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Derivative of:
Derivative of -6/x
Derivative of -2*y
Derivative of ln(x)+x
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Identical expressions
ln(sin((2x+ four)/(2x+ one)))
ln( sinus of ((2x plus 4) divide by (2x plus 1)))
ln( sinus of ((2x plus four) divide by (2x plus one)))
lnsin2x+4/2x+1
ln(sin((2x+4) divide by (2x+1)))
Similar expressions
ln(sin((2x-4)/(2x+1)))
ln(sin((2x+4)/(2x-1)))

The derivative of the function/ ln(sin((2x+4)/(2x+1)))

Derivative of ln(sin((2x+4)/(2x+1)))

The solution

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

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

log(sin((2*x + 4)/(2*x + 1)))

Detail solution

Let $u = \sin{\left(\frac{2 x + 4}{2 x + 1} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\frac{2 x + 4}{2 x + 1} \right)}$ :
1. Let $u = \frac{2 x + 4}{2 x + 1}$ .
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 $\frac{d}{d x} \frac{2 x + 4}{2 x + 1}$ :
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = 2 x + 4$ and $g{\left(x \right)} = 2 x + 1$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Differentiate $2 x + 4$ term by term:
      1. The derivative of the constant $4$ is zero.
      2. 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 is: $2$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Differentiate $2 x + 1$ term by term:
      1. The derivative of the constant $1$ is zero.
      2. 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 is: $2$
    Now plug in to the quotient rule:
    
    $- \frac{6}{\left(2 x + 1\right)^{2}}$
  The result of the chain rule is:
  
  $- \frac{6 \cos{\left(\frac{2 x + 4}{2 x + 1} \right)}}{\left(2 x + 1\right)^{2}}$
The result of the chain rule is:

$- \frac{6 \cos{\left(\frac{2 x + 4}{2 x + 1} \right)}}{\left(2 x + 1\right)^{2} \sin{\left(\frac{2 x + 4}{2 x + 1} \right)}}$
Now simplify:

$- \frac{6}{\left(2 x + 1\right)^{2} \tan{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}}$

The answer is:

$- \frac{6}{\left(2 x + 1\right)^{2} \tan{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}}$

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                   /                                                  2                                  2                                                   \
                   |                     /2*(2 + x)\   /    2*(2 + x)\     3/2*(2 + x)\   /    2*(2 + x)\     /2*(2 + x)\        2/2*(2 + x)\ /    2*(2 + x)\|
                   |                3*cos|---------|   |1 - ---------| *cos |---------|   |1 - ---------| *cos|---------|   3*cos |---------|*|1 - ---------||
   /    2*(2 + x)\ |    6*(2 + x)        \ 1 + 2*x /   \     1 + 2*x /      \ 1 + 2*x /   \     1 + 2*x /     \ 1 + 2*x /         \ 1 + 2*x / \     1 + 2*x /|
16*|1 - ---------|*|3 - --------- + ---------------- + -------------------------------- + ------------------------------- + ---------------------------------|
   \     1 + 2*x / |     1 + 2*x        /2*(2 + x)\               3/2*(2 + x)\                        /2*(2 + x)\                       2/2*(2 + x)\         |
                   |                 sin|---------|            sin |---------|                     sin|---------|                    sin |---------|         |
                   \                    \ 1 + 2*x /                \ 1 + 2*x /                        \ 1 + 2*x /                        \ 1 + 2*x /         /
--------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                   3                                                                          
                                                                          (1 + 2*x)

$$\frac{16 \left(- \frac{2 \left(x + 2\right)}{2 x + 1} + 1\right) \left(- \frac{6 \left(x + 2\right)}{2 x + 1} + \frac{\left(- \frac{2 \left(x + 2\right)}{2 x + 1} + 1\right)^{2} \cos{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}}{\sin{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}} + \frac{\left(- \frac{2 \left(x + 2\right)}{2 x + 1} + 1\right)^{2} \cos^{3}{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}}{\sin^{3}{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}} + \frac{3 \left(- \frac{2 \left(x + 2\right)}{2 x + 1} + 1\right) \cos^{2}{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}}{\sin^{2}{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}} + 3 + \frac{3 \cos{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}}{\sin{\left(\frac{2 \left(x + 2\right)}{2 x + 1} \right)}}\right)}{\left(2 x + 1\right)^{3}}$$

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Derivative of ln(sin((2x+4)/(2x+1)))

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The solution