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

Derivative of x^sqrt(x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   ___
 \/ x 
x
xxx^{\sqrt{x}} 
  /   ___\
d | \/ x |
--\x     /
dx
ddxxx\frac{d}{d x} x^{\sqrt{x}}
Transform
Detail solution

  1. Don't know the steps in finding this derivative.

    But the derivative is

    xx2(log⁡(x)+1)x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right)

  2. Now simplify:

    xx2(log⁡(x)+2)2\frac{x^{\frac{\sqrt{x}}{2}} \left(\log{\left(x \right)} + 2\right)}{2}

The answer is:

xx2(log⁡(x)+2)2\frac{x^{\frac{\sqrt{x}}{2}} \left(\log{\left(x \right)} + 2\right)}{2}

The graph
02468-8-6-4-2-1010-20002000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   ___                  
 \/ x  /  1      log(x)\
x     *|----- + -------|
       |  ___       ___|
       \\/ x    2*\/ x /
xx(log⁡(x)2x+1x)x^{\sqrt{x}} \left(\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right)
Simplify
The second derivative [src] 
   ___ /            2         \
 \/ x  |(2 + log(x))    log(x)|
x     *|------------- - ------|
       |      x           3/2 |
       \                 x    /
-------------------------------
               4
xx((log⁡(x)+2)2x−log⁡(x)x32)4\frac{x^{\sqrt{x}} \left(\frac{\left(\log{\left(x \right)} + 2\right)^{2}}{x} - \frac{\log{\left(x \right)}}{x^{\frac{3}{2}}}\right)}{4}
Simplify
The third derivative [src] 
   ___ /                     3                                   \
 \/ x  |   2     (2 + log(x))    3*log(x)   3*(2 + log(x))*log(x)|
x     *|- ---- + ------------- + -------- - ---------------------|
       |   5/2         3/2          5/2                2         |
       \  x           x            x                  x          /
------------------------------------------------------------------
                                8
xx(−3(log⁡(x)+2)log⁡(x)x2+(log⁡(x)+2)3x32+3log⁡(x)x52−2x52)8\frac{x^{\sqrt{x}} \left(- \frac{3 \left(\log{\left(x \right)} + 2\right) \log{\left(x \right)}}{x^{2}} + \frac{\left(\log{\left(x \right)} + 2\right)^{3}}{x^{\frac{3}{2}}} + \frac{3 \log{\left(x \right)}}{x^{\frac{5}{2}}} - \frac{2}{x^{\frac{5}{2}}}\right)}{8}
Simplify
The graph
Derivative of x^sqrt(x)
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