Derivada de x*exp(-x(((x^2+x+1)^2)^(1/3))*sqrt(x*x+1))

Ha introducido [src]

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   -x*\/   \x  + x + 1/  *\/ x*x + 1 
x*e

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

x*exp(((-x)*((x^2 + x + 1)^2)^(1/3))*sqrt(x*x + 1))

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
$g{\left(x \right)} = e^{- x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = - x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}$ .
2. Derivado $e^{u}$ es.
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} - x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}$ :
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = - x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. Se aplica la regla de la derivada de una multiplicación:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = - x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
      1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        Entonces, como resultado: $-1$
      $g{\left(x \right)} = \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
      1. Sustituimos $u = \left(\left(x^{2} + x\right) + 1\right)^{2}$ .
      2. Según el principio, aplicamos: $\sqrt[3]{u}$ tenemos $\frac{1}{3 u^{\frac{2}{3}}}$
      3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(\left(x^{2} + x\right) + 1\right)^{2}$ :
        
        Sustituimos $u = \left(x^{2} + x\right) + 1$ .
        
        Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
        
        Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(\left(x^{2} + x\right) + 1\right)$ :
        
        diferenciamos $\left(x^{2} + x\right) + 1$ miembro por miembro:
        
        diferenciamos $x^{2} + x$ miembro por miembro:
        
        Según el principio, aplicamos: $x^{2}$ tenemos $2 x$
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        Como resultado de: $2 x + 1$
        
        La derivada de una constante $1$ es igual a cero.
        
        Como resultado de: $2 x + 1$
        
        Como resultado de la secuencia de reglas:
        
        $\left(2 x + 1\right) \left(2 \left(x^{2} + x\right) + 2\right)$
        
        Como resultado de la secuencia de reglas:
        
        $\frac{\left(2 x + 1\right) \left(2 \left(x^{2} + x\right) + 2\right)}{3 \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{4}{3}}}$
      Como resultado de: $- \frac{x \left(2 x + 1\right) \left(2 \left(x^{2} + x\right) + 2\right)}{3 \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{4}{3}}} - \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}$
    $g{\left(x \right)} = \sqrt{x x + 1}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. Sustituimos $u = x x + 1$ .
    2. Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
    3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x x + 1\right)$ :
      1. diferenciamos $x x + 1$ miembro por miembro:
        
        Se aplica la regla de la derivada de una multiplicación:
        
        $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
        
        $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        $g{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        Como resultado de: $2 x$
        
        La derivada de una constante $1$ es igual a cero.
        
        Como resultado de: $2 x$
      Como resultado de la secuencia de reglas:
      
      $\frac{x}{\sqrt{x x + 1}}$
    Como resultado de: $- \frac{x^{2} \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}}{\sqrt{x x + 1}} + \sqrt{x x + 1} \left(- \frac{x \left(2 x + 1\right) \left(2 \left(x^{2} + x\right) + 2\right)}{3 \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{4}{3}}} - \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}\right)$
  Como resultado de la secuencia de reglas:
  
  $\left(- \frac{x^{2} \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}}{\sqrt{x x + 1}} + \sqrt{x x + 1} \left(- \frac{x \left(2 x + 1\right) \left(2 \left(x^{2} + x\right) + 2\right)}{3 \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{4}{3}}} - \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}\right)\right) e^{- x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}}$
Como resultado de: $x \left(- \frac{x^{2} \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}}{\sqrt{x x + 1}} + \sqrt{x x + 1} \left(- \frac{x \left(2 x + 1\right) \left(2 \left(x^{2} + x\right) + 2\right)}{3 \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{4}{3}}} - \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}\right)\right) e^{- x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}} + e^{- x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}}$
Simplificamos:

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

Respuesta:

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

Primera derivada [src]

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  |  _________ |  | 2        |      x*|x  + x + 1|   *(2 + 4*x)|   x *|x  + x + 1|   |  -x*\/   \x  + x + 1/  *\/ x*x + 1     -x*\/   \x  + x + 1/  *\/ x*x + 1 
x*|\/ x*x + 1 *|- |x  + x + 1|    - ---------------------------| - ------------------|*e                                   + e                                  
  |            |                             / 2        \      |        _________    |                                                                          
  \            \                           3*\x  + x + 1/      /      \/ x*x + 1     /

x \left(- \frac{x^{2} \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}}{\sqrt{x x + 1}} + \sqrt{x x + 1} \left(- \frac{x \left(4 x + 2\right) \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}}{3 \left(\left(x^{2} + x\right) + 1\right)} - \left|{\left(x^{2} + x\right) + 1}\right|^{\frac{2}{3}}\right)\right) e^{- x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}} + e^{- x \sqrt[3]{\left(\left(x^{2} + x\right) + 1\right)^{2}} \sqrt{x x + 1}}

Simplificar

Segunda derivada [src]

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-|x*|- |\/  1 + x  *|3 + -------------| + -----------| *|1 + x + x |    + 2*\/  1 + x  *|--------------------------- + ---------------------------- + ------------------- - ------------------------------ + -------------------------------| - -------------------- + -------------------- + --------------------------------------- + -------------------------------| + 6*|1 + x + x |   *|\/  1 + x  *|3 + -------------| + -----------||*e                               
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 \  \  \                                  \/  1 + x  /                                  \                                   \/  |1 + x + x |                                        \1 + x + x /              \1 + x + x /*\/  |1 + x + x | /       \1 + x /               \/  1 + x                        \/  1 + x                    \/  1 + x  *\/  |1 + x + x |  /                     \                                  \/  1 + x  //                                 
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- \frac{\left(x \left(- \frac{9 x^{3} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt{x^{2} + 1} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{3 x \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}} + \frac{18 x \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}} + 2 \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)^{2} \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right) \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{3 x \left(2 x + 1\right)^{2} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + x + 1\right)^{2}} + \frac{6 x \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1} + \frac{3 \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{3 \left(2 x + 1\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1}\right) - \left(\frac{3 x^{2}}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right)\right)^{2} \left|{x^{2} + x + 1}\right|^{\frac{4}{3}}\right) + 6 \left(\frac{3 x^{2}}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right)\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}\right) e^{- x \sqrt{x^{2} + 1} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}}{9}

Simplificar

Tercera derivada [src]

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- \left(- \frac{3 x^{3} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{2 x^{2} \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt{x^{2} + 1} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{x \left(\frac{81 x^{4} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + 1\right)^{\frac{5}{2}}} - \frac{36 x^{3} \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + 1\right)^{\frac{3}{2}} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{36 x^{2} \left(2 x + 1\right)^{2} \delta\left(x^{2} + x + 1\right)}{\sqrt{x^{2} + 1} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{6 x^{2} \left(2 x + 1\right)^{2} \operatorname{sign}^{2}{\left(x^{2} + x + 1 \right)}}{\sqrt{x^{2} + 1} \left|{x^{2} + x + 1}\right|^{\frac{4}{3}}} + \frac{36 x^{2} \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt{x^{2} + 1} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{9 x^{2} \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} - \frac{135 x^{2} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{72 x \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt{x^{2} + 1} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{12 x \left(\frac{2 x \left(2 x + 1\right)^{2} \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right) \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{3 x \left(2 x + 1\right)^{2} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + x + 1\right)^{2}} + \frac{6 x \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1} + \frac{3 \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{3 \left(2 x + 1\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1}\right)}{\sqrt{x^{2} + 1}} + 2 \sqrt{x^{2} + 1} \left(\frac{12 x \left(2 x + 1\right)^{3} \delta\left(x^{2} + x + 1\right)}{\left(x^{2} + x + 1\right) \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{2 x \left(2 x + 1\right)^{3} \operatorname{sign}^{2}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right) \left|{x^{2} + x + 1}\right|^{\frac{4}{3}}} - \frac{12 x \left(2 x + 1\right)^{3} \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right)^{2} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{18 x \left(2 x + 1\right)^{3} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + x + 1\right)^{3}} + \frac{36 x \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right) \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{54 x \left(2 x + 1\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + x + 1\right)^{2}} + \frac{18 \left(2 x + 1\right)^{2} \delta\left(x^{2} + x + 1\right)}{\sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{3 \left(2 x + 1\right)^{2} \operatorname{sign}^{2}{\left(x^{2} + x + 1 \right)}}{\left|{x^{2} + x + 1}\right|^{\frac{4}{3}}} + \frac{12 \left(2 x + 1\right)^{2} \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right) \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{18 \left(2 x + 1\right)^{2} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + x + 1\right)^{2}} + \frac{18 \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{36 \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1}\right) + \left(\frac{3 x^{2}}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right)\right)^{3} \left(x^{2} + x + 1\right)^{2} - 3 \left(\frac{3 x^{2}}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right)\right) \left(- \frac{9 x^{3} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{6 x^{2} \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt{x^{2} + 1} \sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{3 x \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}} + \frac{18 x \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}} + 2 \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)^{2} \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right) \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{3 x \left(2 x + 1\right)^{2} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + x + 1\right)^{2}} + \frac{6 x \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1} + \frac{3 \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{3 \left(2 x + 1\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1}\right)\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}} + \frac{9 \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}} + \frac{54 \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}}\right)}{27} + \frac{x \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}} + \frac{6 x \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\sqrt{x^{2} + 1}} + \frac{2 \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)^{2} \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\left(x^{2} + x + 1\right) \sqrt[3]{\left|{x^{2} + x + 1}\right|}} - \frac{3 x \left(2 x + 1\right)^{2} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{\left(x^{2} + x + 1\right)^{2}} + \frac{6 x \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1} + \frac{3 \left(2 x + 1\right) \operatorname{sign}{\left(x^{2} + x + 1 \right)}}{\sqrt[3]{\left|{x^{2} + x + 1}\right|}} + \frac{3 \left(2 x + 1\right) \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}{x^{2} + x + 1}\right)}{3} - \frac{\left(\frac{3 x^{2}}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} \left(\frac{2 x \left(2 x + 1\right)}{x^{2} + x + 1} + 3\right)\right)^{2} \left|{x^{2} + x + 1}\right|^{\frac{4}{3}}}{3}\right) e^{- x \sqrt{x^{2} + 1} \left|{x^{2} + x + 1}\right|^{\frac{2}{3}}}

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Derivada de xexp(-x(((x^2+x+1)^2)^(1/3))sqrt(x*x+1))

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