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The derivative of the function/ log2(x+1/(2x))

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
   /     1 \
log|x + ---|
   \    2*x/
------------
   log(2)
log⁡(x+12x)log⁡(2)\frac{\log{\left(x + \frac{1}{2 x} \right)}}{\log{\left(2 \right)}} 
log(x + 1/(2*x))/log(2)
Detail solution

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

    1. Let u=x+12xu = x + \frac{1}{2 x}.

    2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

    3. Then, apply the chain rule. Multiply by ddx(x+12x)\frac{d}{d x} \left(x + \frac{1}{2 x}\right):

      1. Differentiate x+12xx + \frac{1}{2 x} term by term:

        1. Apply the power rule: xx goes to 11

        2. Let u=2xu = 2 x.

        3. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

        4. Then, apply the chain rule. Multiply by ddx2x\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: xx goes to 11

            So, the result is: 22

          The result of the chain rule is:

          −12x2- \frac{1}{2 x^{2}}

        The result is: 1−12x21 - \frac{1}{2 x^{2}}

      The result of the chain rule is:

      1−12x2x+12x\frac{1 - \frac{1}{2 x^{2}}}{x + \frac{1}{2 x}}

    So, the result is: 1−12x2(x+12x)log⁡(2)\frac{1 - \frac{1}{2 x^{2}}}{\left(x + \frac{1}{2 x}\right) \log{\left(2 \right)}}

  2. Now simplify:

    2x2−1x(2x2+1)log⁡(2)\frac{2 x^{2} - 1}{x \left(2 x^{2} + 1\right) \log{\left(2 \right)}}

The answer is:

2x2−1x(2x2+1)log⁡(2)\frac{2 x^{2} - 1}{x \left(2 x^{2} + 1\right) \log{\left(2 \right)}}

The graph
02468-8-6-4-2-1010-5050
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       / 1 \    
       |---|    
       \2*x/    
   1 - -----    
         x      
----------------
/     1 \       
|x + ---|*log(2)
\    2*x/
−121xx+1(x+12x)log⁡(2)\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      /
−(2−1x2)22x+1x−2x3(2x+1x)log⁡(2)- \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      /
−2(−(2−1x2)3(2x+1x)2+3(2−1x2)x3(2x+1x)+3x4)(2x+1x)log⁡(2)- \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)}}
Simplify
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