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The derivative of the function/ absolute(3x+1)cbrt(1/(3x)+1)

Derivative of absolute(3x+1)cbrt(1/(3x)+1)

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

You have entered [src] 
              _________
             /  1      
|3*x + 1|*3 /  --- + 1 
          \/   3*x
1+13x3∣3x+1∣\sqrt[3]{1 + \frac{1}{3 x}} \left|{3 x + 1}\right| 
|3*x + 1|*(1/(3*x) + 1)^(1/3)
The graph
02468-8-6-4-2-1010-5050
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The first derivative [src] 
      _________                                  
     /  1                           |3*x + 1|    
3*3 /  --- + 1 *sign(1 + 3*x) - -----------------
  \/   3*x                                    2/3
                                   2 / 1     \   
                                9*x *|--- + 1|   
                                     \3*x    /
31+13x3sign⁡(3x+1)−∣3x+1∣9x2(1+13x)233 \sqrt[3]{1 + \frac{1}{3 x}} \operatorname{sign}{\left(3 x + 1 \right)} - \frac{\left|{3 x + 1}\right|}{9 x^{2} \left(1 + \frac{1}{3 x}\right)^{\frac{2}{3}}}
Simplify
The second derivative [src] 
  /                                                          /        1    \          \
  |                                                          |3 - ---------|*|1 + 3*x||
  |                                                          |      /    1\|          |
  |      _________                                           |    x*|3 + -||          |
  |     /      1                           sign(1 + 3*x)     \      \    x//          |
2*|9*3 /  1 + --- *DiracDelta(1 + 3*x) - ----------------- + -------------------------|
  |  \/       3*x                                      2/3                      2/3   |
  |                                         2 /     1 \              3 /     1 \      |
  |                                      3*x *|1 + ---|          27*x *|1 + ---|      |
  \                                           \    3*x/                \    3*x/      /
2(91+13x3δ(3x+1)−sign⁡(3x+1)3x2(1+13x)23+(3−1x(3+1x))∣3x+1∣27x3(1+13x)23)2 \left(9 \sqrt[3]{1 + \frac{1}{3 x}} \delta\left(3 x + 1\right) - \frac{\operatorname{sign}{\left(3 x + 1 \right)}}{3 x^{2} \left(1 + \frac{1}{3 x}\right)^{\frac{2}{3}}} + \frac{\left(3 - \frac{1}{x \left(3 + \frac{1}{x}\right)}\right) \left|{3 x + 1}\right|}{27 x^{3} \left(1 + \frac{1}{3 x}\right)^{\frac{2}{3}}}\right)
Simplify
The third derivative [src] 
  /                                                                  /         18           5     \                                          \
  |                                                                  |27 - --------- + -----------|*|1 + 3*x|   /        1    \              |
  |                                                                  |       /    1\             2|             |3 - ---------|*sign(1 + 3*x)|
  |                                                                  |     x*|3 + -|    2 /    1\ |             |      /    1\|              |
  |       _________                                                  |       \    x/   x *|3 + -| |             |    x*|3 + -||              |
  |      /      1                            3*DiracDelta(1 + 3*x)   \                    \    x/ /             \      \    x//              |
2*|27*3 /  1 + --- *DiracDelta(1 + 3*x, 1) - --------------------- - ---------------------------------------- + -----------------------------|
  |   \/       3*x                                          2/3                                2/3                                  2/3      |
  |                                              2 /     1 \                        4 /     1 \                          3 /     1 \         |
  |                                             x *|1 + ---|                    81*x *|1 + ---|                       3*x *|1 + ---|         |
  \                                                \    3*x/                          \    3*x/                            \    3*x/         /
2(271+13x3δ(1)(3x+1)−3δ(3x+1)x2(1+13x)23+(3−1x(3+1x))sign⁡(3x+1)3x3(1+13x)23−(27−18x(3+1x)+5x2(3+1x)2)∣3x+1∣81x4(1+13x)23)2 \left(27 \sqrt[3]{1 + \frac{1}{3 x}} \delta^{\left( 1 \right)}\left( 3 x + 1 \right) - \frac{3 \delta\left(3 x + 1\right)}{x^{2} \left(1 + \frac{1}{3 x}\right)^{\frac{2}{3}}} + \frac{\left(3 - \frac{1}{x \left(3 + \frac{1}{x}\right)}\right) \operatorname{sign}{\left(3 x + 1 \right)}}{3 x^{3} \left(1 + \frac{1}{3 x}\right)^{\frac{2}{3}}} - \frac{\left(27 - \frac{18}{x \left(3 + \frac{1}{x}\right)} + \frac{5}{x^{2} \left(3 + \frac{1}{x}\right)^{2}}\right) \left|{3 x + 1}\right|}{81 x^{4} \left(1 + \frac{1}{3 x}\right)^{\frac{2}{3}}}\right)
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