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The derivative of the function/ -lnx/x

Derivative of -lnx/x

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
-log(x) 
--------
   x
(−1)log⁡(x)x\frac{\left(-1\right) \log{\left(x \right)}}{x} 
(-log(x))/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)f{\left(x \right)} = - \log{\left(x \right)} and g(x)=xg{\left(x \right)} = x.

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

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

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

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

The answer is:

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

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  1    log(x)
- -- + ------
   2      2  
  x      x
log⁡(x)x2−1x2\frac{\log{\left(x \right)}}{x^{2}} - \frac{1}{x^{2}}
Simplify
The second derivative [src] 
3 - 2*log(x)
------------
      3     
     x
3−2log⁡(x)x3\frac{3 - 2 \log{\left(x \right)}}{x^{3}}
Simplify
The third derivative [src] 
-11 + 6*log(x)
--------------
       4      
      x
6log⁡(x)−11x4\frac{6 \log{\left(x \right)} - 11}{x^{4}}
Simplify
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