Derivative of lnsin((x-4)/x)

The solution

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   /   /x - 4\\
log|sin|-----||
   \   \  x  //

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

log(sin((x - 4)/x))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/1   x - 4\    /x - 4\
|- - -----|*cos|-----|
|x      2 |    \  x  /
\      x  /           
----------------------
         /x - 4\      
      sin|-----|      
         \  x  /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

               /                                             2                            2                                          \
               |                      /-4 + x\   /    -4 + x\     3/-4 + x\   /    -4 + x\     /-4 + x\        2/-4 + x\ /    -4 + x\|
               |                 3*cos|------|   |1 - ------| *cos |------|   |1 - ------| *cos|------|   3*cos |------|*|1 - ------||
  /    -4 + x\ |    3*(-4 + x)        \  x   /   \      x   /      \  x   /   \      x   /     \  x   /         \  x   / \      x   /|
2*|1 - ------|*|3 - ---------- + ------------- + -------------------------- + ------------------------- + ---------------------------|
  \      x   / |        x            /-4 + x\              3/-4 + x\                    /-4 + x\                     2/-4 + x\       |
               |                  sin|------|           sin |------|                 sin|------|                  sin |------|       |
               \                     \  x   /               \  x   /                    \  x   /                      \  x   /       /
--------------------------------------------------------------------------------------------------------------------------------------
                                                                   3                                                                  
                                                                  x

$$\frac{2 \left(1 - \frac{x - 4}{x}\right) \left(\frac{\left(1 - \frac{x - 4}{x}\right)^{2} \cos{\left(\frac{x - 4}{x} \right)}}{\sin{\left(\frac{x - 4}{x} \right)}} + \frac{\left(1 - \frac{x - 4}{x}\right)^{2} \cos^{3}{\left(\frac{x - 4}{x} \right)}}{\sin^{3}{\left(\frac{x - 4}{x} \right)}} + \frac{3 \left(1 - \frac{x - 4}{x}\right) \cos^{2}{\left(\frac{x - 4}{x} \right)}}{\sin^{2}{\left(\frac{x - 4}{x} \right)}} + 3 + \frac{3 \cos{\left(\frac{x - 4}{x} \right)}}{\sin{\left(\frac{x - 4}{x} \right)}} - \frac{3 \left(x - 4\right)}{x}\right)}{x^{3}}$$

Simplify

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Derivative of lnsin((x-4)/x)

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