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

Solución

Ha introducido [src]

     sin(x)    
---------------
1 + log(sin(x))

$$\frac{\sin{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)} + 1}$$

sin(x)/(1 + log(sin(x)))

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \log{\left(\sin{\left(x \right)} \right)} + 1$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. La derivada del seno es igual al coseno:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $\log{\left(\sin{\left(x \right)} \right)} + 1$ miembro por miembro:
  1. La derivada de una constante $1$ es igual a cero.
  2. Sustituimos $u = \sin{\left(x \right)}$ .
  3. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
  4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de la secuencia de reglas:
    
    $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
  Como resultado de: $\frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \cos{\left(x \right)} - \cos{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right)^{2}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

     cos(x)             cos(x)      
--------------- - ------------------
1 + log(sin(x))                    2
                  (1 + log(sin(x)))

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

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))

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

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))

$$\frac{\left(-1 + \frac{3 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x \right)}}\right)}{\log{\left(\sin{\left(x \right)} \right)} + 1} - \frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{3}{\log{\left(\sin{\left(x \right)} \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right) \sin^{2}{\left(x \right)}} + \frac{3 \cos^{2}{\left(x \right)}}{\left(\log{\left(\sin{\left(x \right)} \right)} + 1\right)^{2} \sin^{2}{\left(x \right)}}\right)}{\log{\left(\sin{\left(x \right)} \right)} + 1} + \frac{3}{\log{\left(\sin{\left(x \right)} \right)} + 1}\right) \cos{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)} + 1}$$

Simplificar

Gráfico

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

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

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