Derivada de y=ln(sqrt((1+sin5x)/(2-cos(3^x))))

Ha introducido [src]

   /     ______________\
   |    / 1 + sin(5*x) |
log|   /  ------------ |
   |  /          / x\  |
   \\/    2 - cos\3 /  /

\log{\left(\sqrt{\frac{\sin{\left(5 x \right)} + 1}{2 - \cos{\left(3^{x} \right)}}} \right)}

log(sqrt((1 + sin(5*x))/(2 - cos(3^x))))

Solución detallada

Sustituimos $u = \sqrt{\frac{\sin{\left(5 x \right)} + 1}{2 - \cos{\left(3^{x} \right)}}}$ .
Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sqrt{\frac{\sin{\left(5 x \right)} + 1}{2 - \cos{\left(3^{x} \right)}}}$ :
1. Sustituimos $u = \frac{\sin{\left(5 x \right)} + 1}{2 - \cos{\left(3^{x} \right)}}$ .
2. Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\sin{\left(5 x \right)} + 1}{2 - \cos{\left(3^{x} \right)}}$ :
  1. Se aplica la regla de la derivada parcial:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(5 x \right)} + 1$ y $g{\left(x \right)} = 2 - \cos{\left(3^{x} \right)}$ .
    
    Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
    1. diferenciamos $\sin{\left(5 x \right)} + 1$ miembro por miembro:
      1. La derivada de una constante $1$ es igual a cero.
      2. Sustituimos $u = 5 x$ .
      3. La derivada del seno es igual al coseno:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} 5 x$ :
        
        La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
        
        Según el principio, aplicamos: $x$ tenemos $1$
        
        Entonces, como resultado: $5$
        
        Como resultado de la secuencia de reglas:
        
        $5 \cos{\left(5 x \right)}$
      Como resultado de: $5 \cos{\left(5 x \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. diferenciamos $2 - \cos{\left(3^{x} \right)}$ miembro por miembro:
      1. La derivada de una constante $2$ es igual a cero.
      2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
        
        Sustituimos $u = 3^{x}$ .
        
        La derivada del coseno es igual a menos el seno:
        
        $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
        
        Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} 3^{x}$ :
        
        $\frac{d}{d x} 3^{x} = 3^{x} \log{\left(3 \right)}$
        
        Como resultado de la secuencia de reglas:
        
        $- 3^{x} \log{\left(3 \right)} \sin{\left(3^{x} \right)}$
        
        Entonces, como resultado: $3^{x} \log{\left(3 \right)} \sin{\left(3^{x} \right)}$
      Como resultado de: $3^{x} \log{\left(3 \right)} \sin{\left(3^{x} \right)}$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{- 3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)} + 5 \left(2 - \cos{\left(3^{x} \right)}\right) \cos{\left(5 x \right)}}{\left(2 - \cos{\left(3^{x} \right)}\right)^{2}}$
  Como resultado de la secuencia de reglas:
  
  $\frac{- 3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)} + 5 \left(2 - \cos{\left(3^{x} \right)}\right) \cos{\left(5 x \right)}}{2 \sqrt{\frac{\sin{\left(5 x \right)} + 1}{2 - \cos{\left(3^{x} \right)}}} \left(2 - \cos{\left(3^{x} \right)}\right)^{2}}$
Como resultado de la secuencia de reglas:

$\frac{\frac{2 - \cos{\left(3^{x} \right)}}{\sin{\left(5 x \right)} + 1} \left(- 3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)} + 5 \left(2 - \cos{\left(3^{x} \right)}\right) \cos{\left(5 x \right)}\right)}{2 \left(2 - \cos{\left(3^{x} \right)}\right)^{2}}$
Simplificamos:

$\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)} + 5 \left(\cos{\left(3^{x} \right)} - 2\right) \cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right) \left(\cos{\left(3^{x} \right)} - 2\right)}$

Respuesta:

$\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)} + 5 \left(\cos{\left(3^{x} \right)} - 2\right) \cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right) \left(\cos{\left(3^{x} \right)} - 2\right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

              /                   x                          / x\\
/       / x\\ |   5*cos(5*x)     3 *(1 + sin(5*x))*log(3)*sin\3 /|
\2 - cos\3 //*|--------------- - --------------------------------|
              |  /       / x\\                          2        |
              |2*\2 - cos\3 //             /       / x\\         |
              \                          2*\2 - cos\3 //         /
------------------------------------------------------------------
                           1 + sin(5*x)

\frac{\left(2 - \cos{\left(3^{x} \right)}\right) \left(- \frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{2 \left(2 - \cos{\left(3^{x} \right)}\right)^{2}} + \frac{5 \cos{\left(5 x \right)}}{2 \left(2 - \cos{\left(3^{x} \right)}\right)}\right)}{\sin{\left(5 x \right)} + 1}

Simplificar

Segunda derivada [src]

                  /              x                          / x\\                                                                                                                                                               /              x                          / x\\               
                  |             3 *(1 + sin(5*x))*log(3)*sin\3 /|                                                                                                                                                             x |             3 *(1 + sin(5*x))*log(3)*sin\3 /|           / x\
                5*|5*cos(5*x) + --------------------------------|*cos(5*x)                                                                                                                                                   3 *|5*cos(5*x) + --------------------------------|*log(3)*sin\3 /
                  |                               / x\          |             2*x    2       2/ x\                   x    2                      / x\    2*x    2                      / x\      x                    / x\      |                               / x\          |               
  25*sin(5*x)     \                       -2 + cos\3 /          /            3   *log (3)*sin \3 /*(1 + sin(5*x))   3 *log (3)*(1 + sin(5*x))*sin\3 /   3   *log (3)*(1 + sin(5*x))*cos\3 /   5*3 *cos(5*x)*log(3)*sin\3 /      \                       -2 + cos\3 /          /               
- ----------- - ---------------------------------------------------------- + ------------------------------------ + --------------------------------- + ----------------------------------- + ---------------------------- - -----------------------------------------------------------------
       2                             2*(1 + sin(5*x))                                                2                         /        / x\\                       /        / x\\                            / x\                                      /        / x\\                        
                                                                                       /        / x\\                        2*\-2 + cos\3 //                     2*\-2 + cos\3 //                    -2 + cos\3 /                                    2*\-2 + cos\3 //                        
                                                                                       \-2 + cos\3 //                                                                                                                                                                                         
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                         1 + sin(5*x)

\frac{\frac{3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \cos{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \sin^{2}{\left(3^{x} \right)}}{\left(\cos{\left(3^{x} \right)} - 2\right)^{2}} - \frac{3^{x} \left(\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + 5 \cos{\left(5 x \right)}\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \sin{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{5 \cdot 3^{x} \log{\left(3 \right)} \sin{\left(3^{x} \right)} \cos{\left(5 x \right)}}{\cos{\left(3^{x} \right)} - 2} - \frac{5 \left(\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + 5 \cos{\left(5 x \right)}\right) \cos{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right)} - \frac{25 \sin{\left(5 x \right)}}{2}}{\sin{\left(5 x \right)} + 1}

Simplificar

Tercera derivada [src]

                   /                x    2                      / x\    2*x    2                      / x\      2*x    2       2/ x\                      x                    / x\\                                                                                                                                                                                /                x    2                      / x\    2*x    2                      / x\      2*x    2       2/ x\                      x                    / x\\                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  
                   |               3 *log (3)*(1 + sin(5*x))*sin\3 /   3   *log (3)*(1 + sin(5*x))*cos\3 /   2*3   *log (3)*sin \3 /*(1 + sin(5*x))   10*3 *cos(5*x)*log(3)*sin\3 /|                         /              x                          / x\\      /              x                          / x\\                                                 x |               3 *log (3)*(1 + sin(5*x))*sin\3 /   3   *log (3)*(1 + sin(5*x))*cos\3 /   2*3   *log (3)*sin \3 /*(1 + sin(5*x))   10*3 *cos(5*x)*log(3)*sin\3 /|           / x\                                                                                                                                                                    /              x                          / x\\                        /              x                          / x\\                                                                                                                                                                                                                  /              x                          / x\\                        
                 5*|-25*sin(5*x) + --------------------------------- + ----------------------------------- + -------------------------------------- + -----------------------------|*cos(5*x)         2      |             3 *(1 + sin(5*x))*log(3)*sin\3 /|      |             3 *(1 + sin(5*x))*log(3)*sin\3 /|                                                3 *|-25*sin(5*x) + --------------------------------- + ----------------------------------- + -------------------------------------- + -----------------------------|*log(3)*sin\3 /                                                                                                                                                          x    2    |             3 *(1 + sin(5*x))*log(3)*sin\3 /|    / x\    2*x    2    |             3 *(1 + sin(5*x))*log(3)*sin\3 /|    / x\                                                                                                                                                                                                        x |             3 *(1 + sin(5*x))*log(3)*sin\3 /|                    / x\
                   |                                  / x\                                 / x\                                       2                                / x\        |            25*cos (5*x)*|5*cos(5*x) + --------------------------------|   25*|5*cos(5*x) + --------------------------------|*sin(5*x)                                          |                                  / x\                                 / x\                                       2                                / x\        |                                                                                                                                                                        3 *log (3)*|5*cos(5*x) + --------------------------------|*sin\3 /   3   *log (3)*|5*cos(5*x) + --------------------------------|*cos\3 /                                                                                                                                                                                                     5*3 *|5*cos(5*x) + --------------------------------|*cos(5*x)*log(3)*sin\3 /
                   |                          -2 + cos\3 /                         -2 + cos\3 /                         /        / x\\                         -2 + cos\3 /        |                         |                               / x\          |      |                               / x\          |             x    3                      / x\      |                          -2 + cos\3 /                         -2 + cos\3 /                         /        / x\\                         -2 + cos\3 /        |                     2*x    3       2/ x\                     3*x    3       3/ x\                      2*x    2       2/ x\                x           / x\                       |                               / x\          |                        |                               / x\          |            3*x    3                      / x\      2*x    3                      / x\       x    2                / x\       2*x    2       / x\               3*x    3                      / x\    / x\        |                               / x\          |                        
  125*cos(5*x)     \                                                                                                    \-2 + cos\3 //                                             /                         \                       -2 + cos\3 /          /      \                       -2 + cos\3 /          /            3 *log (3)*(1 + sin(5*x))*sin\3 /      \                                                                                                    \-2 + cos\3 //                                             /                  3*3   *log (3)*sin \3 /*(1 + sin(5*x))   3*3   *log (3)*sin \3 /*(1 + sin(5*x))   15*3   *log (3)*sin \3 /*cos(5*x)   75*3 *log(3)*sin\3 /*sin(5*x)              \                       -2 + cos\3 /          /                        \                       -2 + cos\3 /          /           3   *log (3)*(1 + sin(5*x))*sin\3 /   3*3   *log (3)*(1 + sin(5*x))*cos\3 /   15*3 *log (3)*cos(5*x)*sin\3 /   15*3   *log (3)*cos\3 /*cos(5*x)   3*3   *log (3)*(1 + sin(5*x))*cos\3 /*sin\3 /        \                       -2 + cos\3 /          /                        
- ------------ - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------ + ----------------------------------------------------------- + --------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------- + -------------------------------------- + --------------------------------- - ----------------------------- - ------------------------------------------------------------------ - -------------------------------------------------------------------- - ----------------------------------- + ------------------------------------- + ------------------------------ + -------------------------------- + --------------------------------------------- + ----------------------------------------------------------------------------
       2                                                                                         1 + sin(5*x)                                                                                                                       2                                                2*(1 + sin(5*x))                                   /        / x\\                                                                                                       / x\                                                                                                               2                                        3                                      2                     /        / x\\                                    /        / x\\                                                        /        / x\\                                         /        / x\\                         /        / x\\                      /        / x\\                    /        / x\\                                        2                                                        /        / x\\                        
                                                                                                                                                                                                                      (1 + sin(5*x))                                                                                                  2*\-2 + cos\3 //                                                                                               -2 + cos\3 /                                                                                                 /        / x\\                           /        / x\\                         /        / x\\                    2*\-2 + cos\3 //                                  2*\-2 + cos\3 //                                                      2*\-2 + cos\3 //                                       2*\-2 + cos\3 //                       2*\-2 + cos\3 //                    2*\-2 + cos\3 //                  2*\-2 + cos\3 //                          /        / x\\                                          (1 + sin(5*x))*\-2 + cos\3 //                        
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  \-2 + cos\3 //                           \-2 + cos\3 //                         \-2 + cos\3 //                                                                                                                                                                                                                                                                                                                                                          \-2 + cos\3 //                                                                                               
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              1 + sin(5*x)

\frac{- \frac{3^{3 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{3} \sin{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{3 \cdot 3^{3 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{3} \sin{\left(3^{x} \right)} \cos{\left(3^{x} \right)}}{\left(\cos{\left(3^{x} \right)} - 2\right)^{2}} + \frac{3 \cdot 3^{3 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{3} \sin^{3}{\left(3^{x} \right)}}{\left(\cos{\left(3^{x} \right)} - 2\right)^{3}} - \frac{3^{2 x} \left(\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + 5 \cos{\left(5 x \right)}\right) \log{\left(3 \right)}^{2} \cos{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{3 \cdot 3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{3} \cos{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{3 \cdot 3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{3} \sin^{2}{\left(3^{x} \right)}}{\left(\cos{\left(3^{x} \right)} - 2\right)^{2}} + \frac{15 \cdot 3^{2 x} \log{\left(3 \right)}^{2} \cos{\left(3^{x} \right)} \cos{\left(5 x \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{15 \cdot 3^{2 x} \log{\left(3 \right)}^{2} \sin^{2}{\left(3^{x} \right)} \cos{\left(5 x \right)}}{\left(\cos{\left(3^{x} \right)} - 2\right)^{2}} - \frac{3^{x} \left(\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + 5 \cos{\left(5 x \right)}\right) \log{\left(3 \right)}^{2} \sin{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{5 \cdot 3^{x} \left(\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + 5 \cos{\left(5 x \right)}\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)} \cos{\left(5 x \right)}}{\left(\sin{\left(5 x \right)} + 1\right) \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{3} \sin{\left(3^{x} \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} - \frac{3^{x} \left(\frac{3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \cos{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + \frac{2 \cdot 3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \sin^{2}{\left(3^{x} \right)}}{\left(\cos{\left(3^{x} \right)} - 2\right)^{2}} + \frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + \frac{10 \cdot 3^{x} \log{\left(3 \right)} \sin{\left(3^{x} \right)} \cos{\left(5 x \right)}}{\cos{\left(3^{x} \right)} - 2} - 25 \sin{\left(5 x \right)}\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} - \frac{75 \cdot 3^{x} \log{\left(3 \right)} \sin{\left(3^{x} \right)} \sin{\left(5 x \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{15 \cdot 3^{x} \log{\left(3 \right)}^{2} \sin{\left(3^{x} \right)} \cos{\left(5 x \right)}}{2 \left(\cos{\left(3^{x} \right)} - 2\right)} + \frac{25 \left(\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + 5 \cos{\left(5 x \right)}\right) \sin{\left(5 x \right)}}{2 \left(\sin{\left(5 x \right)} + 1\right)} + \frac{25 \left(\frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + 5 \cos{\left(5 x \right)}\right) \cos^{2}{\left(5 x \right)}}{\left(\sin{\left(5 x \right)} + 1\right)^{2}} - \frac{125 \cos{\left(5 x \right)}}{2} - \frac{5 \left(\frac{3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \cos{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + \frac{2 \cdot 3^{2 x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \sin^{2}{\left(3^{x} \right)}}{\left(\cos{\left(3^{x} \right)} - 2\right)^{2}} + \frac{3^{x} \left(\sin{\left(5 x \right)} + 1\right) \log{\left(3 \right)}^{2} \sin{\left(3^{x} \right)}}{\cos{\left(3^{x} \right)} - 2} + \frac{10 \cdot 3^{x} \log{\left(3 \right)} \sin{\left(3^{x} \right)} \cos{\left(5 x \right)}}{\cos{\left(3^{x} \right)} - 2} - 25 \sin{\left(5 x \right)}\right) \cos{\left(5 x \right)}}{\sin{\left(5 x \right)} + 1}}{\sin{\left(5 x \right)} + 1}

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Gráfico

Derivada de y=ln(sqrt((1+sin5x)/(2-cos(3^x))))

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Derivada de y=ln(sqrt((1+sin5x)/(2-cos(3^x))))

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