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

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

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

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3 /                2 
\/  (x - 1)*(x + 2)
(x−1)(x+2)23\sqrt[3]{\left(x - 1\right) \left(x + 2\right)^{2}} 
((x - 1)*(x + 2)^2)^(1/3)
Detail solution

  1. Let u=(x−1)(x+2)2u = \left(x - 1\right) \left(x + 2\right)^{2}.

  2. Apply the power rule: u3\sqrt[3]{u} goes to 13u23\frac{1}{3 u^{\frac{2}{3}}}

  3. Then, apply the chain rule. Multiply by ddx(x−1)(x+2)2\frac{d}{d x} \left(x - 1\right) \left(x + 2\right)^{2}:

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=x−1f{\left(x \right)} = x - 1; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Differentiate x−1x - 1 term by term:

        1. Apply the power rule: xx goes to 11

        2. The derivative of the constant −1-1 is zero.

        The result is: 11

      g(x)=(x+2)2g{\left(x \right)} = \left(x + 2\right)^{2}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Let u=x+2u = x + 2.

      2. Apply the power rule: u2u^{2} goes to 2u2 u

      3. Then, apply the chain rule. Multiply by ddx(x+2)\frac{d}{d x} \left(x + 2\right):

        1. Differentiate x+2x + 2 term by term:

          1. Apply the power rule: xx goes to 11

          2. The derivative of the constant 22 is zero.

          The result is: 11

        The result of the chain rule is:

        2x+42 x + 4

      The result is: (x−1)(2x+4)+(x+2)2\left(x - 1\right) \left(2 x + 4\right) + \left(x + 2\right)^{2}

    The result of the chain rule is:

    (x−1)(2x+4)+(x+2)23(x−1)23∣x+2∣43\frac{\left(x - 1\right) \left(2 x + 4\right) + \left(x + 2\right)^{2}}{3 \left(x - 1\right)^{\frac{2}{3}} \left|{x + 2}\right|^{\frac{4}{3}}}

  4. Now simplify:

    x(x+2)(x−1)23∣(x+2)43∣\frac{x \left(x + 2\right)}{\left(x - 1\right)^{\frac{2}{3}} \left|{\left(x + 2\right)^{\frac{4}{3}}}\right|}

The answer is:

x(x+2)(x−1)23∣(x+2)43∣\frac{x \left(x + 2\right)}{\left(x - 1\right)^{\frac{2}{3}} \left|{\left(x + 2\right)^{\frac{4}{3}}}\right|}

The graph
02468-8-6-4-2-1010020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                     /       2                    \
3 _______        2/3 |(x + 2)    (4 + 2*x)*(x - 1)|
\/ x - 1 *|x + 2|   *|-------- + -----------------|
                     \   3               3        /
---------------------------------------------------
                                 2                 
                  (x - 1)*(x + 2)
x−13∣x+2∣23((x−1)(2x+4)3+(x+2)23)(x−1)(x+2)2\frac{\sqrt[3]{x - 1} \left|{x + 2}\right|^{\frac{2}{3}} \left(\frac{\left(x - 1\right) \left(2 x + 4\right)}{3} + \frac{\left(x + 2\right)^{2}}{3}\right)}{\left(x - 1\right) \left(x + 2\right)^{2}}
Simplify
The second derivative [src] 
                   /        2/3     3 ________            \                                             
                   | |2 + x|      2*\/ -1 + x *sign(2 + x)|                                             
                 x*|----------- + ------------------------|                                             
           2/3     |        2/3         3 _________       |                 2/3              2/3        
  x*|2 + x|        \(-1 + x)            \/ |2 + x|        /      2*x*|2 + x|        2*|2 + x|   *(1 + x)
- ------------ + ------------------------------------------ - ------------------- + --------------------
          5/3                    3*(-1 + x)                           2/3                   2/3         
  (-1 + x)                                                    (-1 + x)   *(2 + x)   (-1 + x)   *(2 + x) 
--------------------------------------------------------------------------------------------------------
                                                 2 + x
x(2x−13sign⁡(x+2)∣x+2∣3+∣x+2∣23(x−1)23)3(x−1)−2x∣x+2∣23(x−1)23(x+2)−x∣x+2∣23(x−1)53+2(x+1)∣x+2∣23(x−1)23(x+2)x+2\frac{\frac{x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right)} - \frac{2 x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)} - \frac{x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{5}{3}}} + \frac{2 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)}}{x + 2}
Simplify
3-я производная [src] 
                        /        2/3     3 ________            \       /          2/3   3 ________     2                                      3 ________                  \                                                                                              /        2/3     3 ________            \                                         /        2/3     3 ________            \                                                                                                
                        | |2 + x|      2*\/ -1 + x *sign(2 + x)|       |   |2 + x|      \/ -1 + x *sign (2 + x)        2*sign(2 + x)        6*\/ -1 + x *DiracDelta(2 + x)|                                                                                              | |2 + x|      2*\/ -1 + x *sign(2 + x)|                                         | |2 + x|      2*\/ -1 + x *sign(2 + x)|                                                                                                
                      x*|----------- + ------------------------|   2*x*|- ----------- - ----------------------- + ----------------------- + ------------------------------|                                                                                          2*x*|----------- + ------------------------|                               2*(1 + 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*|2 + x|           \(-1 + x)            \/ |2 + x|        /       \  (-1 + x)             |2 + x|            (-1 + x)   *\/ |2 + x|             \/ |2 + x|           /   5*x*|2 + x|      8*|2 + x|   *(1 + x)      6*x*|2 + x|         10*|2 + x|   *(1 + x)       \(-1 + x)            \/ |2 + x|        /        2*x*sign(2 + x)                  \(-1 + x)            \/ |2 + x|        /      10*x*|2 + x|                  4*x*sign(2 + x)                  4*(1 + x)*sign(2 + x)      
------------------- - ------------------------------------------ + -------------------------------------------------------------------------------------------------------- + -------------- - -------------------- + -------------------- - --------------------- - -------------------------------------------- - ------------------------- + -------------------------------------------------- + --------------------- - --------------------------------- + ---------------------------------
        2/3                                    2                                                                  9*(-1 + x)                                                            8/3            2/3        2           2/3        2             5/3                        3*(-1 + x)*(2 + x)                          5/3 3 _________                   3*(-1 + x)*(2 + x)                             5/3                     2/3         3 _________             2/3         3 _________
(-1 + x)   *(2 + x)                  3*(-1 + x)                                                                                                                               3*(-1 + x)       (-1 + x)   *(2 + x)    (-1 + x)   *(2 + x)    3*(-1 + x)   *(2 + x)                                                  3*(-1 + x)   *\/ |2 + x|                                                         3*(-1 + x)   *(2 + x)   3*(-1 + x)   *(2 + x)*\/ |2 + x|    3*(-1 + x)   *(2 + x)*\/ |2 + x| 
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                                                                                                                                                                                                                                              2 + x
2x(6x−13δ(x+2)∣x+2∣3−x−13sign⁡2(x+2)∣x+2∣43+2sign⁡(x+2)(x−1)23∣x+2∣3−∣x+2∣23(x−1)53)9(x−1)−2x(2x−13sign⁡(x+2)∣x+2∣3+∣x+2∣23(x−1)23)3(x−1)(x+2)−x(2x−13sign⁡(x+2)∣x+2∣3+∣x+2∣23(x−1)23)3(x−1)2−4xsign⁡(x+2)3(x−1)23(x+2)∣x+2∣3+6x∣x+2∣23(x−1)23(x+2)2−2xsign⁡(x+2)3(x−1)53∣x+2∣3+10x∣x+2∣233(x−1)53(x+2)+5x∣x+2∣233(x−1)83+2(x+1)(2x−13sign⁡(x+2)∣x+2∣3+∣x+2∣23(x−1)23)3(x−1)(x+2)+4(x+1)sign⁡(x+2)3(x−1)23(x+2)∣x+2∣3−8(x+1)∣x+2∣23(x−1)23(x+2)2+2∣x+2∣23(x−1)23(x+2)−10(x+1)∣x+2∣233(x−1)53(x+2)x+2\frac{\frac{2 x \left(\frac{6 \sqrt[3]{x - 1} \delta\left(x + 2\right)}{\sqrt[3]{\left|{x + 2}\right|}} - \frac{\sqrt[3]{x - 1} \operatorname{sign}^{2}{\left(x + 2 \right)}}{\left|{x + 2}\right|^{\frac{4}{3}}} + \frac{2 \operatorname{sign}{\left(x + 2 \right)}}{\left(x - 1\right)^{\frac{2}{3}} \sqrt[3]{\left|{x + 2}\right|}} - \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{5}{3}}}\right)}{9 \left(x - 1\right)} - \frac{2 x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} - \frac{x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right)^{2}} - \frac{4 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} + \frac{6 x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} - \frac{2 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{5}{3}} \sqrt[3]{\left|{x + 2}\right|}} + \frac{10 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)} + \frac{5 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{8}{3}}} + \frac{2 \left(x + 1\right) \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} + \frac{4 \left(x + 1\right) \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} - \frac{8 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} + \frac{2 \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)} - \frac{10 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)}}{x + 2}
Simplify
The third derivative [src] 
                        /        2/3     3 ________            \       /          2/3   3 ________     2                                      3 ________                  \                                                                                              /        2/3     3 ________            \                                         /        2/3     3 ________            \                                                                                                
                        | |2 + x|      2*\/ -1 + x *sign(2 + x)|       |   |2 + x|      \/ -1 + x *sign (2 + x)        2*sign(2 + x)        6*\/ -1 + x *DiracDelta(2 + x)|                                                                                              | |2 + x|      2*\/ -1 + x *sign(2 + x)|                                         | |2 + x|      2*\/ -1 + x *sign(2 + x)|                                                                                                
                      x*|----------- + ------------------------|   2*x*|- ----------- - ----------------------- + ----------------------- + ------------------------------|                                                                                          2*x*|----------- + ------------------------|                               2*(1 + 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*|2 + x|           \(-1 + x)            \/ |2 + x|        /       \  (-1 + x)             |2 + x|            (-1 + x)   *\/ |2 + x|             \/ |2 + x|           /   5*x*|2 + x|      8*|2 + x|   *(1 + x)      6*x*|2 + x|         10*|2 + x|   *(1 + x)       \(-1 + x)            \/ |2 + x|        /        2*x*sign(2 + x)                  \(-1 + x)            \/ |2 + x|        /      10*x*|2 + x|                  4*x*sign(2 + x)                  4*(1 + x)*sign(2 + x)      
------------------- - ------------------------------------------ + -------------------------------------------------------------------------------------------------------- + -------------- - -------------------- + -------------------- - --------------------- - -------------------------------------------- - ------------------------- + -------------------------------------------------- + --------------------- - --------------------------------- + ---------------------------------
        2/3                                    2                                                                  9*(-1 + x)                                                            8/3            2/3        2           2/3        2             5/3                        3*(-1 + x)*(2 + x)                          5/3 3 _________                   3*(-1 + x)*(2 + x)                             5/3                     2/3         3 _________             2/3         3 _________
(-1 + x)   *(2 + x)                  3*(-1 + x)                                                                                                                               3*(-1 + x)       (-1 + x)   *(2 + x)    (-1 + x)   *(2 + x)    3*(-1 + x)   *(2 + x)                                                  3*(-1 + x)   *\/ |2 + x|                                                         3*(-1 + x)   *(2 + x)   3*(-1 + x)   *(2 + x)*\/ |2 + x|    3*(-1 + x)   *(2 + x)*\/ |2 + x| 
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                              2 + x
2x(6x−13δ(x+2)∣x+2∣3−x−13sign⁡2(x+2)∣x+2∣43+2sign⁡(x+2)(x−1)23∣x+2∣3−∣x+2∣23(x−1)53)9(x−1)−2x(2x−13sign⁡(x+2)∣x+2∣3+∣x+2∣23(x−1)23)3(x−1)(x+2)−x(2x−13sign⁡(x+2)∣x+2∣3+∣x+2∣23(x−1)23)3(x−1)2−4xsign⁡(x+2)3(x−1)23(x+2)∣x+2∣3+6x∣x+2∣23(x−1)23(x+2)2−2xsign⁡(x+2)3(x−1)53∣x+2∣3+10x∣x+2∣233(x−1)53(x+2)+5x∣x+2∣233(x−1)83+2(x+1)(2x−13sign⁡(x+2)∣x+2∣3+∣x+2∣23(x−1)23)3(x−1)(x+2)+4(x+1)sign⁡(x+2)3(x−1)23(x+2)∣x+2∣3−8(x+1)∣x+2∣23(x−1)23(x+2)2+2∣x+2∣23(x−1)23(x+2)−10(x+1)∣x+2∣233(x−1)53(x+2)x+2\frac{\frac{2 x \left(\frac{6 \sqrt[3]{x - 1} \delta\left(x + 2\right)}{\sqrt[3]{\left|{x + 2}\right|}} - \frac{\sqrt[3]{x - 1} \operatorname{sign}^{2}{\left(x + 2 \right)}}{\left|{x + 2}\right|^{\frac{4}{3}}} + \frac{2 \operatorname{sign}{\left(x + 2 \right)}}{\left(x - 1\right)^{\frac{2}{3}} \sqrt[3]{\left|{x + 2}\right|}} - \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{5}{3}}}\right)}{9 \left(x - 1\right)} - \frac{2 x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} - \frac{x \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right)^{2}} - \frac{4 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} + \frac{6 x \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} - \frac{2 x \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{5}{3}} \sqrt[3]{\left|{x + 2}\right|}} + \frac{10 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)} + \frac{5 x \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{8}{3}}} + \frac{2 \left(x + 1\right) \left(\frac{2 \sqrt[3]{x - 1} \operatorname{sign}{\left(x + 2 \right)}}{\sqrt[3]{\left|{x + 2}\right|}} + \frac{\left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}}}\right)}{3 \left(x - 1\right) \left(x + 2\right)} + \frac{4 \left(x + 1\right) \operatorname{sign}{\left(x + 2 \right)}}{3 \left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right) \sqrt[3]{\left|{x + 2}\right|}} - \frac{8 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)^{2}} + \frac{2 \left|{x + 2}\right|^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{2}{3}} \left(x + 2\right)} - \frac{10 \left(x + 1\right) \left|{x + 2}\right|^{\frac{2}{3}}}{3 \left(x - 1\right)^{\frac{5}{3}} \left(x + 2\right)}}{x + 2}
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