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y=sinx/lnx
Derivada de la función/ x/lnx/ y=sin/ y=sinx/lnx

Derivada de y=sinx/lnx

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

Gráfico:

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Solución

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

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

    1. La derivada del seno es igual al coseno:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

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

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

    Ahora aplicamos la regla de la derivada de una divesión:

    log⁡(x)cos⁡(x)−sin⁡(x)xlog⁡(x)2\frac{\log{\left(x \right)} \cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}}{\log{\left(x \right)}^{2}}

  2. Simplificamos:

    cos⁡(x)log⁡(x)−sin⁡(x)xlog⁡(x)2\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}

Respuesta:

cos⁡(x)log⁡(x)−sin⁡(x)xlog⁡(x)2\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}

Gráfica
02468-8-6-4-2-1010-200100
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
cos(x)     sin(x) 
------ - ---------
log(x)        2   
         x*log (x)
cos⁡(x)log⁡(x)−sin⁡(x)xlog⁡(x)2\frac{\cos{\left(x \right)}}{\log{\left(x \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(x \right)}^{2}}
Simplificar
Segunda derivada [src] 
                     /      2   \       
                     |1 + ------|*sin(x)
          2*cos(x)   \    log(x)/       
-sin(x) - -------- + -------------------
          x*log(x)         2            
                          x *log(x)     
----------------------------------------
                 log(x)
−sin⁡(x)−2cos⁡(x)xlog⁡(x)+(1+2log⁡(x))sin⁡(x)x2log⁡(x)log⁡(x)\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \sin{\left(x \right)}}{x^{2} \log{\left(x \right)}}}{\log{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                       /      3         3   \                               
                     2*|1 + ------ + -------|*sin(x)     /      2   \       
                       |    log(x)      2   |          3*|1 + ------|*cos(x)
          3*sin(x)     \             log (x)/            \    log(x)/       
-cos(x) + -------- - ------------------------------- + ---------------------
          x*log(x)               3                            2             
                                x *log(x)                    x *log(x)      
----------------------------------------------------------------------------
                                   log(x)
−cos⁡(x)+3sin⁡(x)xlog⁡(x)+3(1+2log⁡(x))cos⁡(x)x2log⁡(x)−2(1+3log⁡(x)+3log⁡(x)2)sin⁡(x)x3log⁡(x)log⁡(x)\frac{- \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{x \log{\left(x \right)}} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \cos{\left(x \right)}}{x^{2} \log{\left(x \right)}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \sin{\left(x \right)}}{x^{3} \log{\left(x \right)}}}{\log{\left(x \right)}}
Simplificar
Gráfico
Derivada de y=sinx/lnx
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