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cos(one /log2*x)
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The derivative of the function/ cos(1/log2*x)

Derivative of cos(1/log2*x)

The solution

You have entered [src]

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

$$\cos{\left(\frac{1}{\log{\left(2 x \right)}} \right)}$$

cos(1/log(2*x))

Detail solution

Let $u = \frac{1}{\log{\left(2 x \right)}}$ .
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 $\frac{d}{d x} \frac{1}{\log{\left(2 x \right)}}$ :
1. Let $u = \log{\left(2 x \right)}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(2 x \right)}$ :
  1. Let $u = 2 x$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $\frac{1}{x}$
  The result of the chain rule is:
  
  $- \frac{1}{x \log{\left(2 x \right)}^{2}}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of cos(1/log2*x)

The graph:

Piecewise:

The solution