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

The solution

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   /   /x + 1\\
log|sin|-----||
   \   \  x  //

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

log(sin((x + 1)/x))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/1   x + 1\    /x + 1\
|- - -----|*cos|-----|
|x      2 |    \  x  /
\      x  /           
----------------------
         /x + 1\      
      sin|-----|      
         \  x  /

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

Simplify

The second derivative [src]

             /                 /1 + x\      2/1 + x\ /    1 + x\\ 
             |            2*cos|-----|   cos |-----|*|1 - -----|| 
 /    1 + x\ |    1 + x        \  x  /       \  x  / \      x  /| 
-|1 - -----|*|1 - ----- + ------------ + -----------------------| 
 \      x  / |      x         /1 + x\             2/1 + x\      | 
             |             sin|-----|          sin |-----|      | 
             \                \  x  /              \  x  /      / 
------------------------------------------------------------------
                                 2                                
                                x

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

Simplify

The third derivative [src]

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

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

Simplify

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

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The solution