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log(x)/sqrt(x)
The derivative of the function/ log(x)/sqrt(x)

Derivative of log(x)/sqrt(x)

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

from to

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The solution

You have entered [src] 
log(x)
------
  ___ 
\/ x
log⁡(x)x\frac{\log{\left(x \right)}}{\sqrt{x}} 
d /log(x)\
--|------|
dx|  ___ |
  \\/ x  /
ddxlog⁡(x)x\frac{d}{d x} \frac{\log{\left(x \right)}}{\sqrt{x}}
Transform
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)} = \sqrt{x}.

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

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

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

    1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

    Now plug in to the quotient rule:

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

  2. Now simplify:

    2−log⁡(x)2x32\frac{2 - \log{\left(x \right)}}{2 x^{\frac{3}{2}}}

The answer is:

2−log⁡(x)2x32\frac{2 - \log{\left(x \right)}}{2 x^{\frac{3}{2}}}

The graph
02468-8-6-4-2-1010-100100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   1      log(x)
------- - ------
    ___      3/2
x*\/ x    2*x
1xx−log⁡(x)2x32\frac{1}{\sqrt{x} x} - \frac{\log{\left(x \right)}}{2 x^{\frac{3}{2}}}
Simplify
The second derivative [src] 
     3*log(x)
-2 + --------
        4    
-------------
      5/2    
     x
3log⁡(x)4−2x52\frac{\frac{3 \log{\left(x \right)}}{4} - 2}{x^{\frac{5}{2}}}
Simplify
The third derivative [src] 
46 - 15*log(x)
--------------
       7/2    
    8*x
46−15log⁡(x)8x72\frac{46 - 15 \log{\left(x \right)}}{8 x^{\frac{7}{2}}}
Simplify
The graph
Derivative of log(x)/sqrt(x)
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