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y=(sinx)\(1+log(x)/log(10)*sinx)
Derivada de la función/ y=(sinx)\(1+log(x)/log(10)*sinx)

Derivada de y=(sinx)\(1+log(x)/log(10)*sinx)

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

Ha introducido [src] 
      sin(x)      
------------------
     log(x)       
1 + -------*sin(x)
    log(10)
sin⁡(x)log⁡(x)log⁡(10)sin⁡(x)+1\frac{\sin{\left(x \right)}}{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} \sin{\left(x \right)} + 1} 
sin(x)/(1 + (log(x)/log(10))*sin(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)=log⁡(10)sin⁡(x)f{\left(x \right)} = \log{\left(10 \right)} \sin{\left(x \right)} y g(x)=log⁡(x)sin⁡(x)+log⁡(10)g{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}.

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

    1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

      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)}

      Entonces, como resultado: log⁡(10)cos⁡(x)\log{\left(10 \right)} \cos{\left(x \right)}

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

    1. diferenciamos log⁡(x)sin⁡(x)+log⁡(10)\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)} miembro por miembro:

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

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

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

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

      2. La derivada de una constante log⁡(10)\log{\left(10 \right)} es igual a cero.

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

    Ahora aplicamos la regla de la derivada de una divesión:

    (log⁡(x)sin⁡(x)+log⁡(10))log⁡(10)cos⁡(x)−(log⁡(x)cos⁡(x)+sin⁡(x)x)log⁡(10)sin⁡(x)(log⁡(x)sin⁡(x)+log⁡(10))2\frac{\left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right) \log{\left(10 \right)} \cos{\left(x \right)} - \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \log{\left(10 \right)} \sin{\left(x \right)}}{\left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right)^{2}}

  2. Simplificamos:

    (xlog⁡(10)cos⁡(x)−sin⁡2(x))log⁡(10)x(log⁡(x)sin⁡(x)+log⁡(10))2\frac{\left(x \log{\left(10 \right)} \cos{\left(x \right)} - \sin^{2}{\left(x \right)}\right) \log{\left(10 \right)}}{x \left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right)^{2}}

Respuesta:

(xlog⁡(10)cos⁡(x)−sin⁡2(x))log⁡(10)x(log⁡(x)sin⁡(x)+log⁡(10))2\frac{\left(x \log{\left(10 \right)} \cos{\left(x \right)} - \sin^{2}{\left(x \right)}\right) \log{\left(10 \right)}}{x \left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right)^{2}}

Gráfica
02468-8-6-4-2-1010-1010
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                     /    sin(x)    cos(x)*log(x)\       
                     |- --------- - -------------|*sin(x)
      cos(x)         \  x*log(10)      log(10)   /       
------------------ + ------------------------------------
     log(x)                                     2        
1 + -------*sin(x)          /     log(x)       \         
    log(10)                 |1 + -------*sin(x)|         
                            \    log(10)       /
cos⁡(x)log⁡(x)log⁡(10)sin⁡(x)+1+(−log⁡(x)cos⁡(x)log⁡(10)−sin⁡(x)xlog⁡(10))sin⁡(x)(log⁡(x)log⁡(10)sin⁡(x)+1)2\frac{\cos{\left(x \right)}}{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} \sin{\left(x \right)} + 1} + \frac{\left(- \frac{\log{\left(x \right)} \cos{\left(x \right)}}{\log{\left(10 \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(10 \right)}}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} \sin{\left(x \right)} + 1\right)^{2}}
Simplificar
Segunda derivada [src] 
          /                                                              2\                                           
          |                                      /sin(x)                \ |                                           
          |                                    2*|------ + cos(x)*log(x)| |                                           
          |sin(x)                   2*cos(x)     \  x                   / |                                           
          |------ + log(x)*sin(x) - -------- + ---------------------------|*sin(x)                                    
          |   2                        x       /    log(x)*sin(x)\        |            /sin(x)                \       
          |  x                                 |1 + -------------|*log(10)|          2*|------ + cos(x)*log(x)|*cos(x)
          \                                    \       log(10)   /        /            \  x                   /       
-sin(x) + ------------------------------------------------------------------------ - ---------------------------------
                                /    log(x)*sin(x)\                                     /    log(x)*sin(x)\           
                                |1 + -------------|*log(10)                             |1 + -------------|*log(10)   
                                \       log(10)   /                                     \       log(10)   /           
----------------------------------------------------------------------------------------------------------------------
                                                      log(x)*sin(x)                                                   
                                                  1 + -------------                                                   
                                                         log(10)
−2(log⁡(x)cos⁡(x)+sin⁡(x)x)cos⁡(x)(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)−sin⁡(x)+(2(log⁡(x)cos⁡(x)+sin⁡(x)x)2(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)+log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)sin⁡(x)(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)log⁡(x)sin⁡(x)log⁡(10)+1\frac{- \frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \cos{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \sin{\left(x \right)} + \frac{\left(\frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} + \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}}}{\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1}
Simplificar
Tercera derivada [src] 
          /                                                                             3      /sin(x)                \ /sin(x)                   2*cos(x)\\                                                                                                                        
          |                                                     /sin(x)                \     6*|------ + cos(x)*log(x)|*|------ + log(x)*sin(x) - --------||                                                /                                                              2\       
          |                                                   6*|------ + cos(x)*log(x)|       \  x                   / |   2                        x    ||                                                |                                      /sin(x)                \ |       
          |                 3*sin(x)   3*cos(x)   2*sin(x)      \  x                   /                                \  x                              /|                                                |                                    2*|------ + cos(x)*log(x)| |       
          |-cos(x)*log(x) - -------- - -------- + -------- + ----------------------------- + --------------------------------------------------------------|*sin(x)                                         |sin(x)                   2*cos(x)     \  x                   / |       
          |                    x           2          3                         2                             /    log(x)*sin(x)\                          |                                              3*|------ + log(x)*sin(x) - -------- + ---------------------------|*cos(x)
          |                               x          x       /    log(x)*sin(x)\     2                        |1 + -------------|*log(10)                  |            /sin(x)                \            |   2                        x       /    log(x)*sin(x)\        |       
          |                                                  |1 + -------------| *log (10)                    \       log(10)   /                          |          3*|------ + cos(x)*log(x)|*sin(x)     |  x                                 |1 + -------------|*log(10)|       
          \                                                  \       log(10)   /                                                                           /            \  x                   /            \                                    \       log(10)   /        /       
-cos(x) - --------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------- + --------------------------------------------------------------------------
                                                                         /    log(x)*sin(x)\                                                                             /    log(x)*sin(x)\                                     /    log(x)*sin(x)\                                
                                                                         |1 + -------------|*log(10)                                                                     |1 + -------------|*log(10)                             |1 + -------------|*log(10)                        
                                                                         \       log(10)   /                                                                             \       log(10)   /                                     \       log(10)   /                                
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                     log(x)*sin(x)                                                                                                                                  
                                                                                                                                 1 + -------------                                                                                                                                  
                                                                                                                                        log(10)
3(log⁡(x)cos⁡(x)+sin⁡(x)x)sin⁡(x)(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)−cos⁡(x)+3(2(log⁡(x)cos⁡(x)+sin⁡(x)x)2(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)+log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)cos⁡(x)(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)−(6(log⁡(x)cos⁡(x)+sin⁡(x)x)3(log⁡(x)sin⁡(x)log⁡(10)+1)2log⁡(10)2+6(log⁡(x)cos⁡(x)+sin⁡(x)x)(log⁡(x)sin⁡(x)−2cos⁡(x)x+sin⁡(x)x2)(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)−log⁡(x)cos⁡(x)−3sin⁡(x)x−3cos⁡(x)x2+2sin⁡(x)x3)sin⁡(x)(log⁡(x)sin⁡(x)log⁡(10)+1)log⁡(10)log⁡(x)sin⁡(x)log⁡(10)+1\frac{\frac{3 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \cos{\left(x \right)} + \frac{3 \left(\frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} + \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) \cos{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \frac{\left(\frac{6 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right)^{2} \log{\left(10 \right)}^{2}} + \frac{6 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right)}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \log{\left(x \right)} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{x} - \frac{3 \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3}}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}}}{\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1}
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Gráfico
Derivada de y=(sinx)\(1+log(x)/log(10)*sinx)
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