Sr Examen

Derivada de √x/ln^2x

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

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Definida a trozos:

Solución

Ha introducido [src]
   ___ 
 \/ x  
-------
   2   
log (x)
xlog(x)2\frac{\sqrt{x}}{\log{\left(x \right)}^{2}}
sqrt(x)/log(x)^2
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)=xf{\left(x \right)} = \sqrt{x} y g(x)=log(x)2g{\left(x \right)} = \log{\left(x \right)}^{2}.

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

    1. Según el principio, aplicamos: x\sqrt{x} tenemos 12x\frac{1}{2 \sqrt{x}}

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

    1. Sustituimos u=log(x)u = \log{\left(x \right)}.

    2. Según el principio, aplicamos: u2u^{2} tenemos 2u2 u

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxlog(x)\frac{d}{d x} \log{\left(x \right)}:

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

      Como resultado de la secuencia de reglas:

      2log(x)x\frac{2 \log{\left(x \right)}}{x}

    Ahora aplicamos la regla de la derivada de una divesión:

    log(x)22x2log(x)xlog(x)4\frac{\frac{\log{\left(x \right)}^{2}}{2 \sqrt{x}} - \frac{2 \log{\left(x \right)}}{\sqrt{x}}}{\log{\left(x \right)}^{4}}

  2. Simplificamos:

    log(x)42xlog(x)3\frac{\log{\left(x \right)} - 4}{2 \sqrt{x} \log{\left(x \right)}^{3}}


Respuesta:

log(x)42xlog(x)3\frac{\log{\left(x \right)} - 4}{2 \sqrt{x} \log{\left(x \right)}^{3}}

Gráfica
02468-8-6-4-2-1010-50005000
Primera derivada [src]
       1                2      
--------------- - -------------
    ___    2        ___    3   
2*\/ x *log (x)   \/ x *log (x)
12xlog(x)22xlog(x)3\frac{1}{2 \sqrt{x} \log{\left(x \right)}^{2}} - \frac{2}{\sqrt{x} \log{\left(x \right)}^{3}}
Segunda derivada [src]
                 /      3   \
               2*|1 + ------|
  1     2        \    log(x)/
- - - ------ + --------------
  4   log(x)       log(x)    
-----------------------------
          3/2    2           
         x   *log (x)        
2(1+3log(x))log(x)142log(x)x32log(x)2\frac{\frac{2 \left(1 + \frac{3}{\log{\left(x \right)}}\right)}{\log{\left(x \right)}} - \frac{1}{4} - \frac{2}{\log{\left(x \right)}}}{x^{\frac{3}{2}} \log{\left(x \right)}^{2}}
Tercera derivada [src]
                 /      9         12  \                 
               2*|2 + ------ + -------|     /      3   \
                 |    log(x)      2   |   3*|1 + ------|
3      3         \             log (x)/     \    log(x)/
- + -------- - ------------------------ + --------------
8   2*log(x)            log(x)                log(x)    
--------------------------------------------------------
                       5/2    2                         
                      x   *log (x)                      
3(1+3log(x))log(x)2(2+9log(x)+12log(x)2)log(x)+38+32log(x)x52log(x)2\frac{\frac{3 \left(1 + \frac{3}{\log{\left(x \right)}}\right)}{\log{\left(x \right)}} - \frac{2 \left(2 + \frac{9}{\log{\left(x \right)}} + \frac{12}{\log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)}} + \frac{3}{8} + \frac{3}{2 \log{\left(x \right)}}}{x^{\frac{5}{2}} \log{\left(x \right)}^{2}}
3-я производная [src]
                 /      9         12  \                 
               2*|2 + ------ + -------|     /      3   \
                 |    log(x)      2   |   3*|1 + ------|
3      3         \             log (x)/     \    log(x)/
- + -------- - ------------------------ + --------------
8   2*log(x)            log(x)                log(x)    
--------------------------------------------------------
                       5/2    2                         
                      x   *log (x)                      
3(1+3log(x))log(x)2(2+9log(x)+12log(x)2)log(x)+38+32log(x)x52log(x)2\frac{\frac{3 \left(1 + \frac{3}{\log{\left(x \right)}}\right)}{\log{\left(x \right)}} - \frac{2 \left(2 + \frac{9}{\log{\left(x \right)}} + \frac{12}{\log{\left(x \right)}^{2}}\right)}{\log{\left(x \right)}} + \frac{3}{8} + \frac{3}{2 \log{\left(x \right)}}}{x^{\frac{5}{2}} \log{\left(x \right)}^{2}}
Gráfico
Derivada de √x/ln^2x