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Derivative of (x^5+1)
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Graphing y =:
cbrt(x(x+1)^2)
Identical expressions
cbrt(x(x+ one)^ two)
cubic root of (x(x plus 1) squared )
cubic root of (x(x plus one) to the power of two)
cbrt(x(x+1)2)
cbrtxx+12
cbrt(x(x+1)²)
cbrt(x(x+1) to the power of 2)
cbrtxx+1^2
Similar expressions
cbrt(x(x-1)^2)

The derivative of the function/ cbrt(x(x+1)^2)

Derivative of cbrt(x(x+1)^2)

The solution

You have entered [src]

   ____________
3 /          2 
\/  x*(x + 1)

$$\sqrt[3]{x \left(x + 1\right)^{2}}$$

(x*(x + 1)^2)^(1/3)

Detail solution

Let $u = x \left(x + 1\right)^{2}$ .
Apply the power rule: $\sqrt[3]{u}$ goes to $\frac{1}{3 u^{\frac{2}{3}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \left(x + 1\right)^{2}$ :
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 $1$
  $g{\left(x \right)} = \left(x + 1\right)^{2}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x + 1$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
    1. Differentiate $x + 1$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $1$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $2 x + 2$
  The result is: $x \left(2 x + 2\right) + \left(x + 1\right)^{2}$
The result of the chain rule is:

$\frac{x \left(2 x + 2\right) + \left(x + 1\right)^{2}}{3 x^{\frac{2}{3}} \left|{x + 1}\right|^{\frac{4}{3}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 /       2              \
3 ___        2/3 |(x + 1)    x*(2 + 2*x)|
\/ x *|x + 1|   *|-------- + -----------|
                 \   3            3     /
-----------------------------------------
                         2               
                x*(x + 1)

$$\frac{\sqrt[3]{x} \left|{x + 1}\right|^{\frac{2}{3}} \left(\frac{x \left(2 x + 2\right)}{3} + \frac{\left(x + 1\right)^{2}}{3}\right)}{x \left(x + 1\right)^{2}}$$

Simplify

The second derivative [src]

          /       2/3     3 ___            \                                                                           
          ||1 + x|      2*\/ x *sign(1 + x)|                                                                           
(1 + 3*x)*|---------- + -------------------|                                                                           
          |    2/3          3 _________    |            2/3                      2/3                      2/3          
          \   x             \/ |1 + x|     /   3*|1 + x|   *(1 + 3*x)   6*|1 + x|   *(1 + 3*x)   6*|1 + x|   *(2 + 3*x)
-------------------------------------------- - ---------------------- - ---------------------- + ----------------------
                     x                                   5/3                  2/3                      2/3             
                                                        x                    x   *(1 + x)             x   *(1 + x)     
-----------------------------------------------------------------------------------------------------------------------
                                                       9*(1 + x)

$$\frac{\frac{\left(3 x + 1\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x + 1 \right)}}{\sqrt[3]{\left|{x + 1}\right|}} + \frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{x} - \frac{6 \left(3 x + 1\right) \left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x + 1\right)} + \frac{6 \left(3 x + 2\right) \left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x + 1\right)} - \frac{3 \left(3 x + 1\right) \left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{5}{3}}}}{9 \left(x + 1\right)}$$

Simplify

The third derivative [src]

                         /       2/3     3 ___            \               /         2/3   3 ___     2                               3 ___                  \                                                                                                                    /       2/3     3 ___            \                                         /       2/3     3 ___            \                                                                                    
                         ||1 + x|      2*\/ x *sign(1 + x)|               |  |1 + x|      \/ x *sign (1 + x)    2*sign(1 + x)     6*\/ x *DiracDelta(1 + x)|                                                                                                                    ||1 + x|      2*\/ x *sign(1 + x)|                                         ||1 + x|      2*\/ x *sign(1 + x)|                                                                                    
               (1 + 3*x)*|---------- + -------------------|   2*(1 + 3*x)*|- ---------- - ------------------ + ---------------- + -------------------------|                                                                                                        2*(1 + 3*x)*|---------- + -------------------|                             2*(2 + 3*x)*|---------- + -------------------|                                                                                    
         2/3             |    2/3          3 _________    |               |      5/3                 4/3        2/3 3 _________          3 _________       |            2/3                      2/3                       2/3                      2/3                         |    2/3          3 _________    |                                         |    2/3          3 _________    |             2/3                                                                    
2*|1 + x|                \   x             \/ |1 + x|     /               \     x             |1 + x|          x   *\/ |1 + x|           \/ |1 + x|        /   5*|1 + x|   *(1 + 3*x)   2*|1 + x|   *(1 + 3*x)   10*|1 + x|   *(2 + 3*x)   8*|1 + x|   *(2 + 3*x)               \   x             \/ |1 + x|     /   2*(1 + 3*x)*sign(1 + x)               \   x             \/ |1 + x|     /   10*|1 + x|   *(1 + 3*x)    4*(1 + 3*x)*sign(1 + x)      4*(2 + 3*x)*sign(1 + x)  
------------ - -------------------------------------------- + ---------------------------------------------------------------------------------------------- + ---------------------- + ---------------------- - ----------------------- - ---------------------- - ---------------------------------------------- - ----------------------- + ---------------------------------------------- + ----------------------- - -------------------------- + --------------------------
 2/3                                  2                                                                    27*x                                                           8/3                2/3        2                5/3                     2/3        2                        9*x*(1 + x)                           5/3 3 _________                      9*x*(1 + x)                             5/3                  2/3         3 _________      2/3         3 _________
x   *(1 + x)                       9*x                                                                                                                                 9*x                  x   *(1 + x)              9*x   *(1 + x)          3*x   *(1 + x)                                                            9*x   *\/ |1 + x|                                                            9*x   *(1 + x)       9*x   *(1 + x)*\/ |1 + x|    9*x   *(1 + x)*\/ |1 + x| 
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                      1 + x

$$\frac{\frac{2 \left(3 x + 1\right) \left(\frac{6 \sqrt[3]{x} \delta\left(x + 1\right)}{\sqrt[3]{\left|{x + 1}\right|}} - \frac{\sqrt[3]{x} \operatorname{sign}^{2}{\left(x + 1 \right)}}{\left|{x + 1}\right|^{\frac{4}{3}}} + \frac{2 \operatorname{sign}{\left(x + 1 \right)}}{x^{\frac{2}{3}} \sqrt[3]{\left|{x + 1}\right|}} - \frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{5}{3}}}\right)}{27 x} - \frac{2 \left(3 x + 1\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x + 1 \right)}}{\sqrt[3]{\left|{x + 1}\right|}} + \frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{9 x \left(x + 1\right)} + \frac{2 \left(3 x + 2\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x + 1 \right)}}{\sqrt[3]{\left|{x + 1}\right|}} + \frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{9 x \left(x + 1\right)} - \frac{\left(3 x + 1\right) \left(\frac{2 \sqrt[3]{x} \operatorname{sign}{\left(x + 1 \right)}}{\sqrt[3]{\left|{x + 1}\right|}} + \frac{\left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}}}\right)}{9 x^{2}} - \frac{4 \left(3 x + 1\right) \operatorname{sign}{\left(x + 1 \right)}}{9 x^{\frac{2}{3}} \left(x + 1\right) \sqrt[3]{\left|{x + 1}\right|}} + \frac{4 \left(3 x + 2\right) \operatorname{sign}{\left(x + 1 \right)}}{9 x^{\frac{2}{3}} \left(x + 1\right) \sqrt[3]{\left|{x + 1}\right|}} + \frac{2 \left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x + 1\right)} + \frac{2 \left(3 x + 1\right) \left|{x + 1}\right|^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x + 1\right)^{2}} - \frac{8 \left(3 x + 2\right) \left|{x + 1}\right|^{\frac{2}{3}}}{3 x^{\frac{2}{3}} \left(x + 1\right)^{2}} - \frac{2 \left(3 x + 1\right) \operatorname{sign}{\left(x + 1 \right)}}{9 x^{\frac{5}{3}} \sqrt[3]{\left|{x + 1}\right|}} + \frac{10 \left(3 x + 1\right) \left|{x + 1}\right|^{\frac{2}{3}}}{9 x^{\frac{5}{3}} \left(x + 1\right)} - \frac{10 \left(3 x + 2\right) \left|{x + 1}\right|^{\frac{2}{3}}}{9 x^{\frac{5}{3}} \left(x + 1\right)} + \frac{5 \left(3 x + 1\right) \left|{x + 1}\right|^{\frac{2}{3}}}{9 x^{\frac{8}{3}}}}{x + 1}$$

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Derivative of cbrt(x(x+1)^2)

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