Derivative of ln(sinx/x)

The solution

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   /sin(x)\
log|------|
   \  x   /

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

d /   /sin(x)\\
--|log|------||
dx\   \  x   //

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

Transform

Detail solution

Let $u = \frac{\sin{\left(x \right)}}{x}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\sin{\left(x \right)}}{x}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  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:
  
  $\frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /cos(x)   sin(x)\
x*|------ - ------|
  |  x         2  |
  \           x   /
-------------------
       sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(sinx/x)

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