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The derivative of the function/ sqrt(10*x*lnx)

Derivative of sqrt(10*x*lnx)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  _____________
\/ 10*x*log(x)
$$\sqrt{10 x \log{\left(x \right)}}$$ 
sqrt((10*x)*log(x))
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Apply the product rule:

      ; to find :

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

        1. Apply the power rule: goes to

        So, the result is:

      ; to find :

      1. The derivative of is .

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  _____________               
\/ 10*x*log(x) *(5 + 5*log(x))
------------------------------
         10*x*log(x)
$$\frac{\sqrt{10 x \log{\left(x \right)}} \left(5 \log{\left(x \right)} + 5\right)}{10 x \log{\left(x \right)}}$$
Simplify
The second derivative [src] 
                    /                        2                 \
  ____   __________ |            (1 + log(x))    2*(1 + log(x))|
\/ 10 *\/ x*log(x) *|-2*log(x) + ------------- - --------------|
                    \                log(x)          log(x)    /
----------------------------------------------------------------
                             2                                  
                          4*x *log(x)
$$\frac{\sqrt{10} \sqrt{x \log{\left(x \right)}} \left(\frac{\left(\log{\left(x \right)} + 1\right)^{2}}{\log{\left(x \right)}} - \frac{2 \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}} - 2 \log{\left(x \right)}\right)}{4 x^{2} \log{\left(x \right)}}$$
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))         |
\/ 10 *\/ 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)
$$\frac{\sqrt{10} \sqrt{x \log{\left(x \right)}} \left(\frac{\left(\log{\left(x \right)} + 1\right)^{3}}{8 \log{\left(x \right)}^{2}} - \frac{3 \left(\log{\left(x \right)} + 1\right)^{2}}{4 \log{\left(x \right)}} - \frac{3 \left(\log{\left(x \right)} + 1\right)^{2}}{4 \log{\left(x \right)}^{2}} + \frac{9 \left(\log{\left(x \right)} + 1\right)}{4 \log{\left(x \right)}} + \frac{\log{\left(x \right)} + 1}{\log{\left(x \right)}^{2}} + \log{\left(x \right)} - \frac{1}{2} - \frac{1}{\log{\left(x \right)}}\right)}{x^{3} \log{\left(x \right)}}$$
Simplify
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