Sr Examen

Derivada de x*x^sqrtx

Función f() - derivada -er orden en el punto
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Gráfico:

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Solución

Ha introducido [src]
     ___
   \/ x 
x*x     
xxxx x^{\sqrt{x}}
x*x^(sqrt(x))
Solución detallada
  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)=xxg{\left(x \right)} = x^{\sqrt{x}}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

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

      Perola derivada

      xx2(log(x)+1)x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right)

    Como resultado de: xxx2(log(x)+1)+xxx x^{\frac{\sqrt{x}}{2}} \left(\log{\left(\sqrt{x} \right)} + 1\right) + x^{\sqrt{x}}

  2. Simplificamos:

    xx+xx2+1(log(x)2+1)x^{\sqrt{x}} + x^{\frac{\sqrt{x}}{2} + 1} \left(\frac{\log{\left(x \right)}}{2} + 1\right)


Respuesta:

xx+xx2+1(log(x)2+1)x^{\sqrt{x}} + x^{\frac{\sqrt{x}}{2} + 1} \left(\frac{\log{\left(x \right)}}{2} + 1\right)

Gráfica
02468-8-6-4-2-1010020000
Primera derivada [src]
   ___        ___                  
 \/ x       \/ x  /  1      log(x)\
x      + x*x     *|----- + -------|
                  |  ___       ___|
                  \\/ x    2*\/ x /
xxx(log(x)2x+1x)+xxx x^{\sqrt{x}} \left(\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right) + x^{\sqrt{x}}
Segunda derivada [src]
       /               /            2         \\
       |               |(2 + log(x))    log(x)||
       |             x*|------------- - ------||
   ___ |               |      x           3/2 ||
 \/ x  |2 + log(x)     \                 x    /|
x     *|---------- + --------------------------|
       |    ___                  4             |
       \  \/ x                                 /
xx(x((log(x)+2)2xlog(x)x32)4+log(x)+2x)x^{\sqrt{x}} \left(\frac{x \left(\frac{\left(\log{\left(x \right)} + 2\right)^{2}}{x} - \frac{\log{\left(x \right)}}{x^{\frac{3}{2}}}\right)}{4} + \frac{\log{\left(x \right)} + 2}{\sqrt{x}}\right)
Tercera derivada [src]
   ___ /    /                   3                                   \                            2\
 \/ x  |    | 2     (2 + log(x))    3*log(x)   3*(2 + log(x))*log(x)|   6*log(x)   6*(2 + log(x)) |
x     *|- x*|---- - ------------- - -------- + ---------------------| - -------- + ---------------|
       |    | 5/2         3/2          5/2                2         |      3/2            x       |
       \    \x           x            x                  x          /     x                       /
---------------------------------------------------------------------------------------------------
                                                 8                                                 
xx(x(3(log(x)+2)log(x)x2(log(x)+2)3x323log(x)x52+2x52)+6(log(x)+2)2x6log(x)x32)8\frac{x^{\sqrt{x}} \left(- x \left(\frac{3 \left(\log{\left(x \right)} + 2\right) \log{\left(x \right)}}{x^{2}} - \frac{\left(\log{\left(x \right)} + 2\right)^{3}}{x^{\frac{3}{2}}} - \frac{3 \log{\left(x \right)}}{x^{\frac{5}{2}}} + \frac{2}{x^{\frac{5}{2}}}\right) + \frac{6 \left(\log{\left(x \right)} + 2\right)^{2}}{x} - \frac{6 \log{\left(x \right)}}{x^{\frac{3}{2}}}\right)}{8}
Gráfico
Derivada de x*x^sqrtx