Derivative of cos(ln(x)/x)

The solution

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

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

cos(log(x)/x)

Detail solution

Let $u = \frac{\log{\left(x \right)}}{x}$ .
The derivative of cosine is negative sine:

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

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

 /1    log(x)\    /log(x)\
-|-- - ------|*sin|------|
 | 2      2  |    \  x   /
 \x      x   /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                            3    /log(x)\                                      /log(x)\
                               (-1 + log(x)) *sin|------|   3*(-1 + log(x))*(-3 + 2*log(x))*cos|------|
                    /log(x)\                     \  x   /                                      \  x   /
(-11 + 6*log(x))*sin|------| - -------------------------- + -------------------------------------------
                    \  x   /                2                                    x                     
                                           x                                                           
-------------------------------------------------------------------------------------------------------
                                                    4                                                  
                                                   x

$$\frac{\left(6 \log{\left(x \right)} - 11\right) \sin{\left(\frac{\log{\left(x \right)}}{x} \right)} + \frac{3 \left(\log{\left(x \right)} - 1\right) \left(2 \log{\left(x \right)} - 3\right) \cos{\left(\frac{\log{\left(x \right)}}{x} \right)}}{x} - \frac{\left(\log{\left(x \right)} - 1\right)^{3} \sin{\left(\frac{\log{\left(x \right)}}{x} \right)}}{x^{2}}}{x^{4}}$$

Simplify

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Derivative of cos(ln(x)/x)

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The solution