Sr Examen

Otras calculadoras


y=ln^4((sin^5)2x-x^5)

Derivada de y=ln^4((sin^5)2x-x^5)

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
   4/   5           5\
log \sin (x)*2*x - x /
$$\log{\left(- x^{5} + x 2 \sin^{5}{\left(x \right)} \right)}^{4}$$
log((sin(x)^5*2)*x - x^5)^4
Solución detallada
  1. Sustituimos .

  2. Según el principio, aplicamos: tenemos

  3. Luego se aplica una cadena de reglas. Multiplicamos por :

    1. Sustituimos .

    2. Derivado es .

    3. Luego se aplica una cadena de reglas. Multiplicamos por :

      1. diferenciamos miembro por miembro:

        1. Se aplica la regla de la derivada de una multiplicación:

          ; calculamos :

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

            1. Sustituimos .

            2. Según el principio, aplicamos: tenemos

            3. Luego se aplica una cadena de reglas. Multiplicamos por :

              1. La derivada del seno es igual al coseno:

              Como resultado de la secuencia de reglas:

            Entonces, como resultado:

          ; calculamos :

          1. Según el principio, aplicamos: tenemos

          Como resultado de:

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

          1. Según el principio, aplicamos: tenemos

          Entonces, como resultado:

        Como resultado de:

      Como resultado de la secuencia de reglas:

    Como resultado de la secuencia de reglas:

  4. Simplificamos:


Respuesta:

Gráfica
Primera derivada [src]
     3/   5           5\ /     4      5                4          \
4*log \sin (x)*2*x - x /*\- 5*x  + sin (x)*2 + 10*x*sin (x)*cos(x)/
-------------------------------------------------------------------
                             5           5                         
                          sin (x)*2*x - x                          
$$\frac{4 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right) \log{\left(- x^{5} + x 2 \sin^{5}{\left(x \right)} \right)}^{3}}{- x^{5} + x 2 \sin^{5}{\left(x \right)}}$$
Segunda derivada [src]
                             /                                                                                                                                       2                                             2                          \
                             |                                                                                             /     4        5              4          \    /     4        5              4          \     /  /   4        5   \\|
     2/  /   4        5   \\ |   /   3        5           4                    2       3   \    /  /   4        5   \\   3*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/    \- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/ *log\x*\- x  + 2*sin (x)//|
4*log \x*\- x  + 2*sin (x)//*|10*\2*x  + x*sin (x) - 2*sin (x)*cos(x) - 4*x*cos (x)*sin (x)/*log\x*\- x  + 2*sin (x)// + --------------------------------------------- - ---------------------------------------------------------------------|
                             |                                                                                                           / 4        5   \                                            / 4        5   \                         |
                             \                                                                                                         x*\x  - 2*sin (x)/                                          x*\x  - 2*sin (x)/                         /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                 / 4        5   \                                                                                                              
                                                                                                               x*\x  - 2*sin (x)/                                                                                                              
$$\frac{4 \left(10 \left(2 x^{3} + x \sin^{5}{\left(x \right)} - 4 x \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)} - \frac{\left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{2} \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)} + \frac{3 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{2}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}^{2}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)}$$
3-я производная [src]
  /                                                                                                                                                                 3                                               3                                                                          3                                                                                                                                                                                                                                                                                                       \                          
  |                                                                                                                       /     4        5              4          \      /     4        5              4          \     2/  /   4        5   \\     /     4        5              4          \     /  /   4        5   \\      /     4        5              4          \ /   3        5           4                    2       3   \    /  /   4        5   \\         2/  /   4        5   \\ /     4        5              4          \ /   3        5           4                    2       3   \|                          
  |      2/  /   4        5   \\ /     5         2         2       3              3       2              4          \   6*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/    2*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/ *log \x*\- x  + 2*sin (x)//   9*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/ *log\x*\- x  + 2*sin (x)//   90*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/*\2*x  + x*sin (x) - 2*sin (x)*cos(x) - 4*x*cos (x)*sin (x)/*log\x*\- x  + 2*sin (x)//   30*log \x*\- x  + 2*sin (x)//*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/*\2*x  + x*sin (x) - 2*sin (x)*cos(x) - 4*x*cos (x)*sin (x)/|    /  /   4        5   \\
4*|10*log \x*\- x  + 2*sin (x)//*\3*sin (x) + 6*x  - 12*cos (x)*sin (x) - 12*x*cos (x)*sin (x) + 13*x*sin (x)*cos(x)/ - --------------------------------------------- - ------------------------------------------------------------------------ + ----------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------|*log\x*\- x  + 2*sin (x)//
  |                                                                                                                                                     2                                                            2                                                                          2                                                                                       / 4        5   \                                                                                                                      / 4        5   \                                                         |                          
  |                                                                                                                                   2 / 4        5   \                                           2 / 4        5   \                                                         2 / 4        5   \                                                                                      x*\x  - 2*sin (x)/                                                                                                                    x*\x  - 2*sin (x)/                                                         |                          
  \                                                                                                                                  x *\x  - 2*sin (x)/                                          x *\x  - 2*sin (x)/                                                        x *\x  - 2*sin (x)/                                                                                                                                                                                                                                                                                                       /                          
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                          / 4        5   \                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                        x*\x  - 2*sin (x)/                                                                                                                                                                                                                                                                                                        
$$\frac{4 \left(10 \left(6 x^{2} + 13 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 12 x \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} + 3 \sin^{5}{\left(x \right)} - 12 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}^{2} + \frac{30 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right) \left(2 x^{3} + x \sin^{5}{\left(x \right)} - 4 x \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}^{2}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)} - \frac{90 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right) \left(2 x^{3} + x \sin^{5}{\left(x \right)} - 4 x \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)} - \frac{2 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{3} \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}^{2}}{x^{2} \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)^{2}} + \frac{9 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{3} \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}}{x^{2} \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)^{2}} - \frac{6 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{3}}{x^{2} \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)^{2}}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)}$$
Tercera derivada [src]
  /                                                                                                                                                                 3                                               3                                                                          3                                                                                                                                                                                                                                                                                                       \                          
  |                                                                                                                       /     4        5              4          \      /     4        5              4          \     2/  /   4        5   \\     /     4        5              4          \     /  /   4        5   \\      /     4        5              4          \ /   3        5           4                    2       3   \    /  /   4        5   \\         2/  /   4        5   \\ /     4        5              4          \ /   3        5           4                    2       3   \|                          
  |      2/  /   4        5   \\ /     5         2         2       3              3       2              4          \   6*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/    2*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/ *log \x*\- x  + 2*sin (x)//   9*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/ *log\x*\- x  + 2*sin (x)//   90*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/*\2*x  + x*sin (x) - 2*sin (x)*cos(x) - 4*x*cos (x)*sin (x)/*log\x*\- x  + 2*sin (x)//   30*log \x*\- x  + 2*sin (x)//*\- 5*x  + 2*sin (x) + 10*x*sin (x)*cos(x)/*\2*x  + x*sin (x) - 2*sin (x)*cos(x) - 4*x*cos (x)*sin (x)/|    /  /   4        5   \\
4*|10*log \x*\- x  + 2*sin (x)//*\3*sin (x) + 6*x  - 12*cos (x)*sin (x) - 12*x*cos (x)*sin (x) + 13*x*sin (x)*cos(x)/ - --------------------------------------------- - ------------------------------------------------------------------------ + ----------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------|*log\x*\- x  + 2*sin (x)//
  |                                                                                                                                                     2                                                            2                                                                          2                                                                                       / 4        5   \                                                                                                                      / 4        5   \                                                         |                          
  |                                                                                                                                   2 / 4        5   \                                           2 / 4        5   \                                                         2 / 4        5   \                                                                                      x*\x  - 2*sin (x)/                                                                                                                    x*\x  - 2*sin (x)/                                                         |                          
  \                                                                                                                                  x *\x  - 2*sin (x)/                                          x *\x  - 2*sin (x)/                                                        x *\x  - 2*sin (x)/                                                                                                                                                                                                                                                                                                       /                          
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                                                                                                                                                          / 4        5   \                                                                                                                                                                                                                                                                                                        
                                                                                                                                                                                                                                                                                                        x*\x  - 2*sin (x)/                                                                                                                                                                                                                                                                                                        
$$\frac{4 \left(10 \left(6 x^{2} + 13 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 12 x \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)} + 3 \sin^{5}{\left(x \right)} - 12 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}^{2} + \frac{30 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right) \left(2 x^{3} + x \sin^{5}{\left(x \right)} - 4 x \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}^{2}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)} - \frac{90 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right) \left(2 x^{3} + x \sin^{5}{\left(x \right)} - 4 x \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} - 2 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)} - \frac{2 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{3} \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}^{2}}{x^{2} \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)^{2}} + \frac{9 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{3} \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}}{x^{2} \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)^{2}} - \frac{6 \left(- 5 x^{4} + 10 x \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{5}{\left(x \right)}\right)^{3}}{x^{2} \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)^{2}}\right) \log{\left(x \left(- x^{4} + 2 \sin^{5}{\left(x \right)}\right) \right)}}{x \left(x^{4} - 2 \sin^{5}{\left(x \right)}\right)}$$
Gráfico
Derivada de y=ln^4((sin^5)2x-x^5)