Derivative of lncos((x-1)/x)

The solution

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   /   /x - 1\\
log|cos|-----||
   \   \  x  //

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

log(cos((x - 1)/x))

Detail solution

Let $u = \cos{\left(\frac{x - 1}{x} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(\frac{x - 1}{x} \right)}$ :
1. Let $u = \frac{x - 1}{x}$ .
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 $\frac{d}{d x} \frac{x - 1}{x}$ :
  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:
  
  $- \frac{\sin{\left(\frac{x - 1}{x} \right)}}{x^{2}}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of lncos((x-1)/x)

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