Derivada de y=√((2cos^2x-1)÷(cosx-senx))

Ha introducido [src]

     _________________
    /       2         
   /   2*cos (x) - 1  
  /   --------------- 
\/    cos(x) - sin(x)

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

sqrt((2*cos(x)^2 - 1)/(cos(x) - sin(x)))

Solución detallada

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

$\frac{- \left(- \sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right) - 4 \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{2 \sqrt{\frac{2 \cos^{2}{\left(x \right)} - 1}{- \sin{\left(x \right)} + \cos{\left(x \right)}}} \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}$
Simplificamos:

$\frac{2^{\frac{3}{4}} \left(- 4 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)} + \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(2 x \right)}\right)}{2 \sqrt{\frac{\cos{\left(2 x \right)}}{\cos{\left(x + \frac{\pi}{4} \right)}}} \left(1 - \sin{\left(2 x \right)}\right)}$

Respuesta:

$\frac{2^{\frac{3}{4}} \left(- 4 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(x + \frac{\pi}{4} \right)} + \sin{\left(x + \frac{\pi}{4} \right)} \cos{\left(2 x \right)}\right)}{2 \sqrt{\frac{\cos{\left(2 x \right)}}{\cos{\left(x + \frac{\pi}{4} \right)}}} \left(1 - \sin{\left(2 x \right)}\right)}$

Primera derivada [src]

     _________________                                                                        
    /       2          //     2       \                                    \                  
   /   2*cos (x) - 1   |\2*cos (x) - 1/*(cos(x) + sin(x))   2*cos(x)*sin(x)|                  
  /   --------------- *|--------------------------------- - ---------------|*(cos(x) - sin(x))
\/    cos(x) - sin(x)  |                          2         cos(x) - sin(x)|                  
                       \       2*(cos(x) - sin(x))                         /                  
----------------------------------------------------------------------------------------------
                                             2                                                
                                        2*cos (x) - 1

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

Simplificar

Segunda derivada [src]

                          /                                                                                  2                                                                                                                                                                                                                              \
                          |                            /                  /          2   \                  \                                          /                  /          2   \                  \                       /                  /          2   \                  \                                                  |
     ____________________ |                            |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                          |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                       |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                                  |
    /  /          2   \   |                            |4*cos(x)*sin(x) + ----------------------------------|                     2 /          2   \   |4*cos(x)*sin(x) + ----------------------------------|*(cos(x) + sin(x))   2*|4*cos(x)*sin(x) + ----------------------------------|*cos(x)*sin(x)                                    |
   /  -\-1 + 2*cos (x)/   |  1      2           2      \                           -cos(x) + sin(x)         /    (cos(x) + sin(x)) *\-1 + 2*cos (x)/   \                           -cos(x) + sin(x)         /                       \                           -cos(x) + sin(x)         /                 4*(cos(x) + sin(x))*cos(x)*sin(x)|
  /   ------------------ *|- - - cos (x) + 2*sin (x) + ------------------------------------------------------- + ----------------------------------- - ------------------------------------------------------------------------ - ---------------------------------------------------------------------- + ---------------------------------|
\/     -cos(x) + sin(x)   |  2                                              /          2   \                                               2                                     2*(-cos(x) + sin(x))                                                                   2                                           -cos(x) + sin(x)        |
                          \                                               4*\-1 + 2*cos (x)/                             (-cos(x) + sin(x))                                                                                                                   -1 + 2*cos (x)                                                                /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                          2                                                                                                                                                                  
                                                                                                                                                                -1 + 2*cos (x)

\frac{\sqrt{- \frac{2 \cos^{2}{\left(x \right)} - 1}{\sin{\left(x \right)} - \cos{\left(x \right)}}} \left(\frac{\left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{2}}{4 \left(2 \cos^{2}{\left(x \right)} - 1\right)} - \frac{2 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{\left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)} + 2 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} - \frac{1}{2} + \frac{4 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right)}{2 \cos^{2}{\left(x \right)} - 1}

Simplificar

Tercera derivada [src]

                          /                                                                        3                     /                                                2 /          2   \                                    \                                                                                                                                                                                                                                                                                      /                  /          2   \                  \ /                                                2 /          2   \                                    \                                                                                                                                                                             2                   /                                                2 /          2   \                                    \                                                                         2                                                                                                             \
                          |                  /                  /          2   \                  \                      |          2           2      2*(cos(x) + sin(x)) *\-1 + 2*cos (x)/   8*(cos(x) + sin(x))*cos(x)*sin(x)|                                                                                   /                  /          2   \                  \                                                    /                  /          2   \                  \                                   |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))| |          2           2      2*(cos(x) + sin(x)) *\-1 + 2*cos (x)/   8*(cos(x) + sin(x))*cos(x)*sin(x)|                      /                  /          2   \                  \                                           /                  /          2   \                  \                    |          2           2      2*(cos(x) + sin(x)) *\-1 + 2*cos (x)/   8*(cos(x) + sin(x))*cos(x)*sin(x)|                   /                  /          2   \                  \                        /                  /          2   \                  \                                |
     ____________________ |                  |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|    (cos(x) + sin(x))*|-1 - 2*cos (x) + 4*sin (x) + ------------------------------------- + ---------------------------------|                                                                              2    |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                               2    |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                 3*|4*cos(x)*sin(x) + ----------------------------------|*|-1 - 2*cos (x) + 4*sin (x) + ------------------------------------- + ---------------------------------|         2       2    |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                           |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                  4*|-1 - 2*cos (x) + 4*sin (x) + ------------------------------------- + ---------------------------------|*cos(x)*sin(x)     |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                        |                  \-1 + 2*cos (x)/*(cos(x) + sin(x))|                                |
    /  /          2   \   |                  |4*cos(x)*sin(x) + ----------------------------------|                      |                                                        2                     -cos(x) + sin(x)        |        2                                           3 /          2   \   2*cos (x)*|4*cos(x)*sin(x) + ----------------------------------|     /          2   \                     2*sin (x)*|4*cos(x)*sin(x) + ----------------------------------|        2                          \                           -cos(x) + sin(x)         / |                                                        2                     -cos(x) + sin(x)        |   16*cos (x)*sin (x)*|4*cos(x)*sin(x) + ----------------------------------|                       2                 3*|4*cos(x)*sin(x) + ----------------------------------| *cos(x)*sin(x)     |                                                        2                     -cos(x) + sin(x)        |                 3*|4*cos(x)*sin(x) + ----------------------------------| *(cos(x) + sin(x))   4*|4*cos(x)*sin(x) + ----------------------------------|*(cos(x) + sin(x))*cos(x)*sin(x)|
   /  -\-1 + 2*cos (x)/   |                  \                           -cos(x) + sin(x)         /                      \                                      (-cos(x) + sin(x))                                              /   6*sin (x)*(cos(x) + sin(x))   3*(cos(x) + sin(x)) *\-1 + 2*cos (x)/             \                           -cos(x) + sin(x)         /   2*\-1 + 2*cos (x)/*(cos(x) + sin(x))             \                           -cos(x) + sin(x)         /   6*cos (x)*(cos(x) + sin(x))                                                            \                                      (-cos(x) + sin(x))                                              /                      \                           -cos(x) + sin(x)         /   12*(cos(x) + sin(x)) *cos(x)*sin(x)     \                           -cos(x) + sin(x)         /                    \                                      (-cos(x) + sin(x))                                              /                   \                           -cos(x) + sin(x)         /                        \                           -cos(x) + sin(x)         /                                |
  /   ------------------ *|4*cos(x)*sin(x) - ------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------- - --------------------------- - ------------------------------------- - ---------------------------------------------------------------- - ------------------------------------ + ---------------------------------------------------------------- + --------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------- - ----------------------------------- + ----------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------|
\/     -cos(x) + sin(x)   |                                                      2                                                                          -cos(x) + sin(x)                                                              -cos(x) + sin(x)                                   3                                               2                                         -cos(x) + sin(x)                                                2                                     -cos(x) + sin(x)                                                                                  /          2   \                                                                                                                      2                                                         2                                                      2                                                                                             2                                                                                /          2   \                                                                   /          2   \                                              |
                          |                                      /          2   \                                                                                                                                                                                          (-cos(x) + sin(x))                                      -1 + 2*cos (x)                                                                                            -1 + 2*cos (x)                                                                                                                                  4*\-1 + 2*cos (x)/                                                                                                      /          2   \                                        (-cos(x) + sin(x))                                       /          2   \                                                                                    -1 + 2*cos (x)                                                                           4*\-1 + 2*cos (x)/*(-cos(x) + sin(x))                                                \-1 + 2*cos (x)/*(-cos(x) + sin(x))                           |
                          \                                    8*\-1 + 2*cos (x)/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    \-1 + 2*cos (x)/                                                                                                 \-1 + 2*cos (x)/                                                                                                                                                                                                                                                                                                                                /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     -1 + 2*cos (x)

\frac{\sqrt{- \frac{2 \cos^{2}{\left(x \right)} - 1}{\sin{\left(x \right)} - \cos{\left(x \right)}}} \left(- \frac{\left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{3}}{8 \left(2 \cos^{2}{\left(x \right)} - 1\right)^{2}} + \frac{3 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}}{\left(2 \cos^{2}{\left(x \right)} - 1\right)^{2}} + \frac{3 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)^{2} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{4 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)} - \frac{3 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \left(4 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - 1 + \frac{8 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right)}{4 \left(2 \cos^{2}{\left(x \right)} - 1\right)} + \frac{2 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \sin^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{2 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{16 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(2 \cos^{2}{\left(x \right)} - 1\right)^{2}} - \frac{4 \left(4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)} + 4 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{4 \left(4 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - 1 + \frac{8 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 1} - \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \left(4 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)} - 1 + \frac{8 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}\right)}{\sin{\left(x \right)} - \cos{\left(x \right)}} - \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} + \frac{6 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} - \frac{12 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}} - \frac{3 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{3} \left(2 \cos^{2}{\left(x \right)} - 1\right)}{\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{3}}\right)}{2 \cos^{2}{\left(x \right)} - 1}

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Gráfico

Derivada de y=√((2cos^2x-1)÷(cosx-senx))

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Derivada de y=√((2cos^2x-1)÷(cosx-senx))

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