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Derivada de:
Derivada de y=8t³+3t²+2
Derivada de y=(×³+1)^10
Derivada de y=cx^2
Derivada de y=2*3,14*n*t
Expresiones idénticas
y=ln(sqrt((uno +sin5x)/(dos -cos3^x)))
y es igual a ln( raíz cuadrada de ((1 más seno de 5x) dividir por (2 menos coseno de 3 en el grado x)))
y es igual a ln( raíz cuadrada de ((uno más seno de 5x) dividir por (dos menos coseno de 3 en el grado x)))
y=ln(√((1+sin5x)/(2-cos3^x)))
y=ln(sqrt((1+sin5x)/(2-cos3x)))
y=lnsqrt1+sin5x/2-cos3x
y=lnsqrt1+sin5x/2-cos3^x
y=ln(sqrt((1+sin5x) dividir por (2-cos3^x)))
Expresiones semejantes
y=ln(sqrt((1+sin5x)/(2+cos3^x)))
y=ln(sqrt((1-sin5x)/(2-cos3^x)))

Derivada de la función/ y=ln(sqrt((1+sin5x)/(2-cos3^x)))

Derivada de y=ln(sqrt((1+sin5x)/(2-cos3^x)))

Solución

Ha introducido [src]

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

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

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

Solución detallada

Sustituimos $u = \sqrt{\frac{\sin{\left(5 x \right)} + 1}{2 - \cos^{x}{\left(3 \right)}}}$ .
Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sqrt{\frac{\sin{\left(5 x \right)} + 1}{2 - \cos^{x}{\left(3 \right)}}}$ :
1. Sustituimos $u = \frac{\sin{\left(5 x \right)} + 1}{2 - \cos^{x}{\left(3 \right)}}$ .
2. Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\sin{\left(5 x \right)} + 1}{2 - \cos^{x}{\left(3 \right)}}$ :
  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 $g{\left(x \right)} = 2 - \cos^{x}{\left(3 \right)}$ .
    
    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 $2 - \cos^{x}{\left(3 \right)}$ 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.
        
        $\frac{d}{d x} \cos^{x}{\left(3 \right)} = \left(\log{\left(- \cos{\left(3 \right)} \right)} + i \pi\right) \cos^{x}{\left(3 \right)}$
        
        Entonces, como resultado: $- \left(\log{\left(- \cos{\left(3 \right)} \right)} + i \pi\right) \cos^{x}{\left(3 \right)}$
      Como resultado de: $- \left(\log{\left(- \cos{\left(3 \right)} \right)} + i \pi\right) \cos^{x}{\left(3 \right)}$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{5 \left(2 - \cos^{x}{\left(3 \right)}\right) \cos{\left(5 x \right)} + \left(\log{\left(- \cos{\left(3 \right)} \right)} + i \pi\right) \left(\sin{\left(5 x \right)} + 1\right) \cos^{x}{\left(3 \right)}}{\left(2 - \cos^{x}{\left(3 \right)}\right)^{2}}$
  Como resultado de la secuencia de reglas:
  
  $\frac{5 \left(2 - \cos^{x}{\left(3 \right)}\right) \cos{\left(5 x \right)} + \left(\log{\left(- \cos{\left(3 \right)} \right)} + i \pi\right) \left(\sin{\left(5 x \right)} + 1\right) \cos^{x}{\left(3 \right)}}{2 \sqrt{\frac{\sin{\left(5 x \right)} + 1}{2 - \cos^{x}{\left(3 \right)}}} \left(2 - \cos^{x}{\left(3 \right)}\right)^{2}}$
Como resultado de la secuencia de reglas:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

Derivada de y=ln(sqrt((1+sin5x)/(2-cos3^x)))

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Derivada de y=ln(sqrt((1+sin5x)/(2-cos3^x)))

Gráfico:

Definida a trozos:

Solución