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The derivative of the function/ ln^2x-(ln(ln(x)))

Derivative of ln^2x-(ln(ln(x)))

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

You have entered [src] 
   2                 
log (x) - log(log(x))
log⁡(x)2−log⁡(log⁡(x))\log{\left(x \right)}^{2} - \log{\left(\log{\left(x \right)} \right)} 
log(x)^2 - log(log(x))
Detail solution

  1. Differentiate log⁡(x)2−log⁡(log⁡(x))\log{\left(x \right)}^{2} - \log{\left(\log{\left(x \right)} \right)} term by term:

    1. Let u=log⁡(x)u = \log{\left(x \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddxlog⁡(x)\frac{d}{d x} \log{\left(x \right)}:

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

      The result of the chain rule is:

      2log⁡(x)x\frac{2 \log{\left(x \right)}}{x}

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

      1. Let u=log⁡(x)u = \log{\left(x \right)}.

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

      3. Then, apply the chain rule. Multiply by ddxlog⁡(x)\frac{d}{d x} \log{\left(x \right)}:

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

        The result of the chain rule is:

        1xlog⁡(x)\frac{1}{x \log{\left(x \right)}}

      So, the result is: −1xlog⁡(x)- \frac{1}{x \log{\left(x \right)}}

    The result is: 2log⁡(x)x−1xlog⁡(x)\frac{2 \log{\left(x \right)}}{x} - \frac{1}{x \log{\left(x \right)}}

  2. Now simplify:

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

The answer is:

2log⁡(x)2−1xlog⁡(x)\frac{2 \log{\left(x \right)}^{2} - 1}{x \log{\left(x \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*log(x)
- -------- + --------
  x*log(x)      x
2log⁡(x)x−1xlog⁡(x)\frac{2 \log{\left(x \right)}}{x} - \frac{1}{x \log{\left(x \right)}}
Simplify
The second derivative [src] 
      1         1              
2 + ------ + ------- - 2*log(x)
    log(x)      2              
             log (x)           
-------------------------------
                2              
               x
−2log⁡(x)+2+1log⁡(x)+1log⁡(x)2x2\frac{- 2 \log{\left(x \right)} + 2 + \frac{1}{\log{\left(x \right)}} + \frac{1}{\log{\left(x \right)}^{2}}}{x^{2}}
Simplify
The third derivative [src] 
        3        2         2              
-6 - ------- - ------ - ------- + 4*log(x)
        2      log(x)      3              
     log (x)            log (x)           
------------------------------------------
                     3                    
                    x
4log⁡(x)−6−2log⁡(x)−3log⁡(x)2−2log⁡(x)3x3\frac{4 \log{\left(x \right)} - 6 - \frac{2}{\log{\left(x \right)}} - \frac{3}{\log{\left(x \right)}^{2}} - \frac{2}{\log{\left(x \right)}^{3}}}{x^{3}}
Simplify
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