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Derivative of y=cos((1)/log2*x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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
  1. Let .

  2. The derivative of cosine is negative sine:

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. Apply the power rule: goes to

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. The derivative of is .

      3. Then, apply the chain rule. Multiply by :

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result of the chain rule is:

      The result of the chain rule is:

    The result of the chain rule is:


The answer is:

The graph
The first derivative [src]
   /   1    \
sin|--------|
   \log(2*x)/
-------------
      2      
 x*log (2*x) 
$$\frac{\sin{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{x \log{\left(2 x \right)}^{2}}$$
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)                   
$$- \frac{\sin{\left(\frac{1}{\log{\left(2 x \right)}} \right)} + \frac{2 \sin{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{\log{\left(2 x \right)}} + \frac{\cos{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{\log{\left(2 x \right)}^{2}}}{x^{2} \log{\left(2 x \right)}^{2}}$$
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)                                             
$$\frac{2 \sin{\left(\frac{1}{\log{\left(2 x \right)}} \right)} + \frac{6 \sin{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{\log{\left(2 x \right)}} + \frac{6 \sin{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{\log{\left(2 x \right)}^{2}} + \frac{3 \cos{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{\log{\left(2 x \right)}^{2}} + \frac{6 \cos{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{\log{\left(2 x \right)}^{3}} - \frac{\sin{\left(\frac{1}{\log{\left(2 x \right)}} \right)}}{\log{\left(2 x \right)}^{4}}}{x^{3} \log{\left(2 x \right)}^{2}}$$