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y=(sinx)\(1+lnsinx)
Derivada de la función/ lnsinx/ (sinx)/ y=(sinx)\(1+lnsinx)

Derivada de y=(sinx)\(1+lnsinx)

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

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

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

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

      2. Sustituimos u=sin⁡(x)u = \sin{\left(x \right)}.

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

      4. Luego se aplica una cadena de reglas. Multiplicamos por ddxsin⁡(x)\frac{d}{d x} \sin{\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 la secuencia de reglas:

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

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

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

    log⁡(sin⁡(x))cos⁡(x)(log⁡(sin⁡(x))+1)2\frac{\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right)^{2}}

Respuesta:

log⁡(sin⁡(x))cos⁡(x)(log⁡(sin⁡(x))+1)2\frac{\log{\left(\sin{\left(x \right)} \right)} \cos{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right)^{2}}

Gráfica
02468-8-6-4-2-101020000-10000
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Primera derivada [src] 
     cos(x)             cos(x)      
--------------- - ------------------
1 + log(sin(x))                    2
                  (1 + log(sin(x)))
cos⁡(x)log⁡(sin⁡(x))+1−cos⁡(x)(log⁡(sin⁡(x))+1)2\frac{\cos{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)} + 1} - \frac{\cos{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right)^{2}}
Simplificar
Segunda derivada [src] 
          /       2                   2           \                                  
          |    cos (x)           2*cos (x)        |                                  
          |1 + ------- + -------------------------|*sin(x)                           
          |       2                           2   |                      2           
          \    sin (x)   (1 + log(sin(x)))*sin (x)/                 2*cos (x)        
-sin(x) + ------------------------------------------------ - ------------------------
                          1 + log(sin(x))                    (1 + log(sin(x)))*sin(x)
-------------------------------------------------------------------------------------
                                   1 + log(sin(x))
−sin⁡(x)+(1+cos⁡2(x)sin⁡2(x)+2cos⁡2(x)(log⁡(sin⁡(x))+1)sin⁡2(x))sin⁡(x)log⁡(sin⁡(x))+1−2cos⁡2(x)(log⁡(sin⁡(x))+1)sin⁡(x)log⁡(sin⁡(x))+1\frac{- \sin{\left(x \right)} + \frac{\left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x \right)}}\right) \sin{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)} + 1} - \frac{2 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin{\left(x \right)}}}{\log{\left(\sin{\left(x \right)} \right)} + 1}
Simplificar
Tercera derivada [src] 
/                         /                         2                   2                           2            \     /       2                   2           \\       
|                         |           3          cos (x)           3*cos (x)                   3*cos (x)         |     |    cos (x)           2*cos (x)        ||       
|                       2*|1 + --------------- + ------- + ------------------------- + --------------------------|   3*|1 + ------- + -------------------------||       
|                         |    1 + log(sin(x))      2                           2                       2    2   |     |       2                           2   ||       
|            3            \                      sin (x)   (1 + log(sin(x)))*sin (x)   (1 + log(sin(x))) *sin (x)/     \    sin (x)   (1 + log(sin(x)))*sin (x)/|       
|-1 + --------------- - ------------------------------------------------------------------------------------------ + -------------------------------------------|*cos(x)
\     1 + log(sin(x))                                        1 + log(sin(x))                                                       1 + log(sin(x))              /       
------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                            1 + log(sin(x))
(−1+3(1+cos⁡2(x)sin⁡2(x)+2cos⁡2(x)(log⁡(sin⁡(x))+1)sin⁡2(x))log⁡(sin⁡(x))+1−2(1+cos⁡2(x)sin⁡2(x)+3log⁡(sin⁡(x))+1+3cos⁡2(x)(log⁡(sin⁡(x))+1)sin⁡2(x)+3cos⁡2(x)(log⁡(sin⁡(x))+1)2sin⁡2(x))log⁡(sin⁡(x))+1+3log⁡(sin⁡(x))+1)cos⁡(x)log⁡(sin⁡(x))+1\frac{\left(-1 + \frac{3 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x \right)}}\right)}{\log{\left(\sin{\left(x \right)} \right)} + 1} - \frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{3}{\log{\left(\sin{\left(x \right)} \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x \right)}} + \frac{3 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right)^{2} \sin^{2}{\left(x \right)}}\right)}{\log{\left(\sin{\left(x \right)} \right)} + 1} + \frac{3}{\log{\left(\sin{\left(x \right)} \right)} + 1}\right) \cos{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)} + 1}
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Gráfico
Derivada de y=(sinx)\(1+lnsinx)
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