Derivative of sqrt(xlnx)

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  __________
\/ x*log(x)

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

sqrt(x*log(x))

Detail solution

Let $u = x \log{\left(x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \log{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\log{\left(x \right)} + 1$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  __________ /1   log(x)\
\/ x*log(x) *|- + ------|
             \2     2   /
-------------------------
         x*log(x)

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

Simplify

The second derivative [src]

             /                        2                 \
  __________ |            (1 + log(x))    2*(1 + log(x))|
\/ x*log(x) *|-2*log(x) + ------------- - --------------|
             \                log(x)          log(x)    /
---------------------------------------------------------
                          2                              
                       4*x *log(x)

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

Simplify

The third derivative [src]

             /                                          2                 2               3                          \
  __________ |  1     1      1 + log(x)   3*(1 + log(x))    3*(1 + log(x))    (1 + log(x))    9*(1 + log(x))         |
\/ x*log(x) *|- - - ------ + ---------- - --------------- - --------------- + ------------- + -------------- + log(x)|
             |  2   log(x)       2            4*log(x)              2                2           4*log(x)            |
             \                log (x)                          4*log (x)        8*log (x)                            /
----------------------------------------------------------------------------------------------------------------------
                                                       3                                                              
                                                      x *log(x)

$$\frac{\sqrt{x \log{\left(x \right)}} \left(\frac{\left(\log{\left(x \right)} + 1\right)^{3}}{8 \log{\left(x \right)}^{2}} - \frac{3 \left(\log{\left(x \right)} + 1\right)^{2}}{4 \log{\left(x \right)}} - \frac{3 \left(\log{\left(x \right)} + 1\right)^{2}}{4 \log{\left(x \right)}^{2}} + \frac{9 \left(\log{\left(x \right)} + 1\right)}{4 \log{\left(x \right)}} + \frac{\log{\left(x \right)} + 1}{\log{\left(x \right)}^{2}} + \log{\left(x \right)} - \frac{1}{2} - \frac{1}{\log{\left(x \right)}}\right)}{x^{3} \log{\left(x \right)}}$$

Simplify

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Derivative of sqrt(xlnx)

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The solution