Derivative of sqrt(x)/ln(x)

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  ___ 
\/ x  
------
log(x)

$$\frac{\sqrt{x}}{\log{\left(x \right)}}$$

sqrt(x)/log(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt{x}$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

$\frac{\frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}}{\log{\left(x \right)}^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1                1      
-------------- - -------------
    ___            ___    2   
2*\/ x *log(x)   \/ x *log (x)

$$\frac{1}{2 \sqrt{x} \log{\left(x \right)}} - \frac{1}{\sqrt{x} \log{\left(x \right)}^{2}}$$

Simplify

The second derivative [src]

                     2   
               1 + ------
  1     1          log(x)
- - - ------ + ----------
  4   log(x)     log(x)  
-------------------------
        3/2              
       x   *log(x)

$$\frac{\frac{1 + \frac{2}{\log{\left(x \right)}}}{\log{\left(x \right)}} - \frac{1}{4} - \frac{1}{\log{\left(x \right)}}}{x^{\frac{3}{2}} \log{\left(x \right)}}$$

Simplify

The third derivative [src]

                 /      3         3   \                 
               2*|1 + ------ + -------|     /      2   \
                 |    log(x)      2   |   3*|1 + ------|
3      3         \             log (x)/     \    log(x)/
- + -------- - ------------------------ + --------------
8   4*log(x)            log(x)               2*log(x)   
--------------------------------------------------------
                       5/2                              
                      x   *log(x)

$$\frac{\frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right)}{2 \log{\left(x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)}} + \frac{3}{8} + \frac{3}{4 \log{\left(x \right)}}}{x^{\frac{5}{2}} \log{\left(x \right)}}$$

Simplify

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Derivative of sqrt(x)/ln(x)

The graph:

Piecewise:

The solution