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(x^(x+1))/(x+1)
Derivada de la función/ (x+1)/ (x^(x+1))/(x+1)

Derivada de (x^(x+1))/(x+1)

Función f() - derivada -er orden en el punto
v

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
 x + 1
x     
------
x + 1
xx+1x+1\frac{x^{x + 1}}{x + 1} 
x^(x + 1)/(x + 1)
Solución detallada

  1. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=xx+1f{\left(x \right)} = x^{x + 1} y g(x)=x+1g{\left(x \right)} = x + 1.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. No logro encontrar los pasos en la búsqueda de esta derivada.

      Perola derivada

      (x+1)x+1(log⁡(x+1)+1)\left(x + 1\right)^{x + 1} \left(\log{\left(x + 1 \right)} + 1\right)

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. diferenciamos x+1x + 1 miembro por miembro:

      1. La derivada de una constante 11 es igual a cero.

      2. Según el principio, aplicamos: xx tenemos 11

      Como resultado de: 11

    Ahora aplicamos la regla de la derivada de una divesión:

    −xx+1+(x+1)(x+1)x+1(log⁡(x+1)+1)(x+1)2\frac{- x^{x + 1} + \left(x + 1\right) \left(x + 1\right)^{x + 1} \left(\log{\left(x + 1 \right)} + 1\right)}{\left(x + 1\right)^{2}}

  2. Simplificamos:

    −xx+1+(x+1)x+2(log⁡(x+1)+1)(x+1)2\frac{- x^{x + 1} + \left(x + 1\right)^{x + 2} \left(\log{\left(x + 1 \right)} + 1\right)}{\left(x + 1\right)^{2}}

Respuesta:

−xx+1+(x+1)x+2(log⁡(x+1)+1)(x+1)2\frac{- x^{x + 1} + \left(x + 1\right)^{x + 2} \left(\log{\left(x + 1 \right)} + 1\right)}{\left(x + 1\right)^{2}}

Gráfica
02468-8-6-4-2-1010050000000000
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Trazar el gráfico f'(x)
Primera derivada [src] 
              x + 1 /x + 1         \
    x + 1    x     *|----- + log(x)|
   x                \  x           /
- -------- + -----------------------
         2            x + 1         
  (x + 1)
xx+1(log⁡(x)+x+1x)x+1−xx+1(x+1)2\frac{x^{x + 1} \left(\log{\left(x \right)} + \frac{x + 1}{x}\right)}{x + 1} - \frac{x^{x + 1}}{\left(x + 1\right)^{2}}
Simplificar
Segunda derivada [src] 
       /                                   1 + x     /1 + x         \\
       |                2              2 - -----   2*|----- + log(x)||
 1 + x |/1 + x         \       2             x       \  x           /|
x     *||----- + log(x)|  + -------- + --------- - ------------------|
       |\  x           /           2       x             1 + x       |
       \                    (1 + x)                                  /
----------------------------------------------------------------------
                                1 + x
xx+1((log⁡(x)+x+1x)2−2(log⁡(x)+x+1x)x+1+2(x+1)2+2−x+1xx)x+1\frac{x^{x + 1} \left(\left(\log{\left(x \right)} + \frac{x + 1}{x}\right)^{2} - \frac{2 \left(\log{\left(x \right)} + \frac{x + 1}{x}\right)}{x + 1} + \frac{2}{\left(x + 1\right)^{2}} + \frac{2 - \frac{x + 1}{x}}{x}\right)}{x + 1}
Simplificar
Tercera derivada [src] 
       /                                                 /                        1 + x\                                                      \
       |                                                 |                2   2 - -----|                                                      |
       |                                   2*(1 + x)     |/1 + x         \          x  |     /1 + x         \     /    1 + x\ /1 + x         \|
       |                3              3 - ---------   3*||----- + log(x)|  + ---------|   6*|----- + log(x)|   3*|2 - -----|*|----- + log(x)||
 1 + x |/1 + x         \       6               x         \\  x           /        x    /     \  x           /     \      x  / \  x           /|
x     *||----- + log(x)|  - -------- - ------------- - --------------------------------- + ------------------ + ------------------------------|
       |\  x           /           3          2                      1 + x                             2                      x               |
       \                    (1 + x)          x                                                  (1 + x)                                       /
-----------------------------------------------------------------------------------------------------------------------------------------------
                                                                     1 + x
xx+1((log⁡(x)+x+1x)3−3((log⁡(x)+x+1x)2+2−x+1xx)x+1+6(log⁡(x)+x+1x)(x+1)2−6(x+1)3+3(2−x+1x)(log⁡(x)+x+1x)x−3−2(x+1)xx2)x+1\frac{x^{x + 1} \left(\left(\log{\left(x \right)} + \frac{x + 1}{x}\right)^{3} - \frac{3 \left(\left(\log{\left(x \right)} + \frac{x + 1}{x}\right)^{2} + \frac{2 - \frac{x + 1}{x}}{x}\right)}{x + 1} + \frac{6 \left(\log{\left(x \right)} + \frac{x + 1}{x}\right)}{\left(x + 1\right)^{2}} - \frac{6}{\left(x + 1\right)^{3}} + \frac{3 \left(2 - \frac{x + 1}{x}\right) \left(\log{\left(x \right)} + \frac{x + 1}{x}\right)}{x} - \frac{3 - \frac{2 \left(x + 1\right)}{x}}{x^{2}}\right)}{x + 1}
Simplificar
Gráfico
Derivada de (x^(x+1))/(x+1)
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