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(x×lnx)/((x+1)×(lnx+1))
Derivada de la función/ x×lnx/ (x+1)/ lnx+1/ (x×lnx)/((x+1)×(lnx+1))

Derivada de (x×lnx)/((x+1)×(lnx+1))

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
      x*log(x)      
--------------------
(x + 1)*(log(x) + 1)
xlog⁡(x)(x+1)(log⁡(x)+1)\frac{x \log{\left(x \right)}}{\left(x + 1\right) \left(\log{\left(x \right)} + 1\right)} 
(x*log(x))/(((x + 1)*(log(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)=xlog⁡(x)f{\left(x \right)} = x \log{\left(x \right)} y g(x)=(x+1)(log⁡(x)+1)g{\left(x \right)} = \left(x + 1\right) \left(\log{\left(x \right)} + 1\right).

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

    1. Se aplica la regla de la derivada de una multiplicación:

      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)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: xx tenemos 11

      g(x)=log⁡(x)g{\left(x \right)} = \log{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

      Como resultado de: log⁡(x)+1\log{\left(x \right)} + 1

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

    1. Se aplica la regla de la derivada de una multiplicación:

      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; calculamos ddxf(x)\frac{d}{d x} f{\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

      g(x)=log⁡(x)+1g{\left(x \right)} = \log{\left(x \right)} + 1; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. diferenciamos log⁡(x)+1\log{\left(x \right)} + 1 miembro por miembro:

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

        2. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

        Como resultado de: 1x\frac{1}{x}

      Como resultado de: log⁡(x)+1+x+1x\log{\left(x \right)} + 1 + \frac{x + 1}{x}

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-2525
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                                      /              x + 1\       
                                    x*|-1 - log(x) - -----|*log(x)
         1                            \                x  /       
--------------------*(1 + log(x)) + ------------------------------
(x + 1)*(log(x) + 1)                           2             2    
                                        (x + 1) *(log(x) + 1)
x(−log⁡(x)−1−x+1x)log⁡(x)(x+1)2(log⁡(x)+1)2+1(x+1)(log⁡(x)+1)(log⁡(x)+1)\frac{x \left(- \log{\left(x \right)} - 1 - \frac{x + 1}{x}\right) \log{\left(x \right)}}{\left(x + 1\right)^{2} \left(\log{\left(x \right)} + 1\right)^{2}} + \frac{1}{\left(x + 1\right) \left(\log{\left(x \right)} + 1\right)} \left(\log{\left(x \right)} + 1\right)
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Segunda derivada [src] 
                               /    1 + x                                                                1 + x       1 + x         \       
                               |1 + ----- + log(x)                                                   2 - -----   1 + ----- + log(x)|       
      /    1 + x         \     |      x              /  1           1       \ /    1 + x         \         x           x           |       
    2*|1 + ----- + log(x)|   x*|------------------ + |----- + --------------|*|1 + ----- + log(x)| - --------- + ------------------|*log(x)
1     \      x           /     \      1 + x          \1 + x   x*(1 + log(x))/ \      x           /       x         x*(1 + log(x))  /       
- - ---------------------- + --------------------------------------------------------------------------------------------------------------
x           1 + x                                                         (1 + x)*(1 + log(x))                                             
-------------------------------------------------------------------------------------------------------------------------------------------
                                                            (1 + x)*(1 + log(x))
x((1x+1+1x(log⁡(x)+1))(log⁡(x)+1+x+1x)+log⁡(x)+1+x+1xx+1−2−x+1xx+log⁡(x)+1+x+1xx(log⁡(x)+1))log⁡(x)(x+1)(log⁡(x)+1)−2(log⁡(x)+1+x+1x)x+1+1x(x+1)(log⁡(x)+1)\frac{\frac{x \left(\left(\frac{1}{x + 1} + \frac{1}{x \left(\log{\left(x \right)} + 1\right)}\right) \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right) + \frac{\log{\left(x \right)} + 1 + \frac{x + 1}{x}}{x + 1} - \frac{2 - \frac{x + 1}{x}}{x} + \frac{\log{\left(x \right)} + 1 + \frac{x + 1}{x}}{x \left(\log{\left(x \right)} + 1\right)}\right) \log{\left(x \right)}}{\left(x + 1\right) \left(\log{\left(x \right)} + 1\right)} - \frac{2 \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right)}{x + 1} + \frac{1}{x}}{\left(x + 1\right) \left(\log{\left(x \right)} + 1\right)}
Simplificar
Tercera derivada [src] 
                                                                                                                                            /                                                                                                    2*(1 + x)     /    1 + x         \       1 + x            /  1           1       \ /    1 + x         \   /    1 + x\ /  1           1       \     /    1 + x\      /    1 + x\      /    1 + x         \   /  1           1       \ /    1 + x         \     /    1 + x         \\       
         /    1 + x                                                                1 + x       1 + x         \                              |                                                                                                3 - ---------   3*|1 + ----- + log(x)|   1 + ----- + log(x)   |----- + --------------|*|1 + ----- + log(x)|   |2 - -----|*|----- + --------------|   3*|2 - -----|    3*|2 - -----|    3*|1 + ----- + log(x)|   |----- + --------------|*|1 + ----- + log(x)|   4*|1 + ----- + log(x)||       
         |1 + ----- + log(x)                                                   2 - -----   1 + ----- + log(x)|                              |/    1 + x         \ /   2              1                 2                     2           \           x         \      x           /         x              \1 + x   x*(1 + log(x))/ \      x           /   \      x  / \1 + x   x*(1 + log(x))/     \      x  /      \      x  /      \      x           /   \1 + x   x*(1 + log(x))/ \      x           /     \      x           /|       
         |      x              /  1           1       \ /    1 + x         \         x           x           |     /    1 + x         \   x*||1 + ----- + log(x)|*|-------- + --------------- + ---------------- + ----------------------| - ------------- + ---------------------- + ------------------ + --------------------------------------------- - ------------------------------------ - ------------- - --------------- + ---------------------- + --------------------------------------------- + ----------------------|*log(x)
       3*|------------------ + |----- + --------------|*|1 + ----- + log(x)| - --------- + ------------------|   3*|1 + ----- + log(x)|     |\      x           / |       2    2                 2             2   x*(1 + x)*(1 + log(x))|          2                      2            2                                      1 + x                                        x                       x*(1 + x)      2                    2             2                      x*(1 + log(x))                  x*(1 + x)*(1 + log(x))|       
  1      \      1 + x          \1 + x   x*(1 + log(x))/ \      x           /       x         x*(1 + log(x))  /     \      x           /     \                     \(1 + x)    x *(1 + log(x))   x *(1 + log(x))                          /         x                (1 + x)            x *(1 + log(x))                                                                                                            x *(1 + log(x))      x *(1 + log(x))                                                                             /       
- -- + ------------------------------------------------------------------------------------------------------- - ---------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   2                                                    1 + x                                                    x*(1 + x)*(1 + log(x))                                                                                                                                                                                          (1 + x)*(1 + log(x))                                                                                                                                                                                      
  x                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                            (1 + x)*(1 + log(x))
−x((log⁡(x)+1+x+1x)(2(x+1)2+2x(x+1)(log⁡(x)+1)+1x2(log⁡(x)+1)+2x2(log⁡(x)+1)2)+(1x+1+1x(log⁡(x)+1))(log⁡(x)+1+x+1x)x+1+3(log⁡(x)+1+x+1x)(x+1)2−(2−x+1x)(1x+1+1x(log⁡(x)+1))x−3(2−x+1x)x(x+1)+(1x+1+1x(log⁡(x)+1))(log⁡(x)+1+x+1x)x(log⁡(x)+1)+4(log⁡(x)+1+x+1x)x(x+1)(log⁡(x)+1)−3(2−x+1x)x2(log⁡(x)+1)−3−2(x+1)xx2+log⁡(x)+1+x+1xx2(log⁡(x)+1)+3(log⁡(x)+1+x+1x)x2(log⁡(x)+1)2)log⁡(x)(x+1)(log⁡(x)+1)+3((1x+1+1x(log⁡(x)+1))(log⁡(x)+1+x+1x)+log⁡(x)+1+x+1xx+1−2−x+1xx+log⁡(x)+1+x+1xx(log⁡(x)+1))x+1−3(log⁡(x)+1+x+1x)x(x+1)(log⁡(x)+1)−1x2(x+1)(log⁡(x)+1)\frac{- \frac{x \left(\left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right) \left(\frac{2}{\left(x + 1\right)^{2}} + \frac{2}{x \left(x + 1\right) \left(\log{\left(x \right)} + 1\right)} + \frac{1}{x^{2} \left(\log{\left(x \right)} + 1\right)} + \frac{2}{x^{2} \left(\log{\left(x \right)} + 1\right)^{2}}\right) + \frac{\left(\frac{1}{x + 1} + \frac{1}{x \left(\log{\left(x \right)} + 1\right)}\right) \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right)}{x + 1} + \frac{3 \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right)}{\left(x + 1\right)^{2}} - \frac{\left(2 - \frac{x + 1}{x}\right) \left(\frac{1}{x + 1} + \frac{1}{x \left(\log{\left(x \right)} + 1\right)}\right)}{x} - \frac{3 \left(2 - \frac{x + 1}{x}\right)}{x \left(x + 1\right)} + \frac{\left(\frac{1}{x + 1} + \frac{1}{x \left(\log{\left(x \right)} + 1\right)}\right) \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right)}{x \left(\log{\left(x \right)} + 1\right)} + \frac{4 \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right)}{x \left(x + 1\right) \left(\log{\left(x \right)} + 1\right)} - \frac{3 \left(2 - \frac{x + 1}{x}\right)}{x^{2} \left(\log{\left(x \right)} + 1\right)} - \frac{3 - \frac{2 \left(x + 1\right)}{x}}{x^{2}} + \frac{\log{\left(x \right)} + 1 + \frac{x + 1}{x}}{x^{2} \left(\log{\left(x \right)} + 1\right)} + \frac{3 \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right)}{x^{2} \left(\log{\left(x \right)} + 1\right)^{2}}\right) \log{\left(x \right)}}{\left(x + 1\right) \left(\log{\left(x \right)} + 1\right)} + \frac{3 \left(\left(\frac{1}{x + 1} + \frac{1}{x \left(\log{\left(x \right)} + 1\right)}\right) \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right) + \frac{\log{\left(x \right)} + 1 + \frac{x + 1}{x}}{x + 1} - \frac{2 - \frac{x + 1}{x}}{x} + \frac{\log{\left(x \right)} + 1 + \frac{x + 1}{x}}{x \left(\log{\left(x \right)} + 1\right)}\right)}{x + 1} - \frac{3 \left(\log{\left(x \right)} + 1 + \frac{x + 1}{x}\right)}{x \left(x + 1\right) \left(\log{\left(x \right)} + 1\right)} - \frac{1}{x^{2}}}{\left(x + 1\right) \left(\log{\left(x \right)} + 1\right)}
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Gráfico
Derivada de (x×lnx)/((x+1)×(lnx+1))
    © Sr Examen – Калькулятор