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log(x)^2/x

Derivative of log(x)^2/x

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

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Piecewise:

The solution

You have entered [src]
   2   
log (x)
-------
   x   
log(x)2x\frac{\log{\left(x \right)}^{2}}{x}
log(x)^2/x
Detail solution
  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=log(x)2f{\left(x \right)} = \log{\left(x \right)}^{2} and g(x)=xg{\left(x \right)} = x.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    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}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    Now plug in to the quotient rule:

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

  2. Now simplify:

    (2log(x))log(x)x2\frac{\left(2 - \log{\left(x \right)}\right) \log{\left(x \right)}}{x^{2}}


The answer is:

(2log(x))log(x)x2\frac{\left(2 - \log{\left(x \right)}\right) \log{\left(x \right)}}{x^{2}}

The graph
02468-8-6-4-2-1010-10001000
The first derivative [src]
     2              
  log (x)   2*log(x)
- ------- + --------
      2         2   
     x         x    
log(x)2x2+2log(x)x2- \frac{\log{\left(x \right)}^{2}}{x^{2}} + \frac{2 \log{\left(x \right)}}{x^{2}}
The second derivative [src]
  /       2              \
2*\1 + log (x) - 3*log(x)/
--------------------------
             3            
            x             
2(log(x)23log(x)+1)x3\frac{2 \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x^{3}}
The third derivative [src]
  /          2               \
2*\-6 - 3*log (x) + 11*log(x)/
------------------------------
               4              
              x               
2(3log(x)2+11log(x)6)x4\frac{2 \left(- 3 \log{\left(x \right)}^{2} + 11 \log{\left(x \right)} - 6\right)}{x^{4}}
The graph
Derivative of log(x)^2/x