Derivative of log2(x+1/(2x))

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

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

log(x + 1/(2*x))/log(2)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = x + \frac{1}{2 x}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + \frac{1}{2 x}\right)$ :
  1. Differentiate $x + \frac{1}{2 x}$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. Let $u = 2 x$ .
    3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
    4. 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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
      The result of the chain rule is:
      
      $- \frac{1}{2 x^{2}}$
    The result is: $1 - \frac{1}{2 x^{2}}$
  The result of the chain rule is:
  
  $\frac{1 - \frac{1}{2 x^{2}}}{x + \frac{1}{2 x}}$
So, the result is: $\frac{1 - \frac{1}{2 x^{2}}}{\left(x + \frac{1}{2 x}\right) \log{\left(2 \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

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Derivative of log2(x+1/(2x))

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