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Expresiones idénticas
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xsin(cos(x)^(i))
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xsincosx^i
Expresiones semejantes
x*sin(cosx^(i))

Derivada de la función/ x*sin/ cos(x)/ x*sin(cos(x)^(i))

Derivada de x*sin(cos(x)^(i))

Solución

Ha introducido [src]

     /   I   \
x*sin\cos (x)/

$$x \sin{\left(\cos^{i}{\left(x \right)} \right)}$$

x*sin(cos(x)^i)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Según el principio, aplicamos: $x$ tenemos $1$
$g{\left(x \right)} = \sin{\left(\cos^{i}{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = \cos^{i}{\left(x \right)}$ .
2. La derivada del seno es igual al coseno:
  
  $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos^{i}{\left(x \right)}$ :
  1. Sustituimos $u = \cos{\left(x \right)}$ .
  2. Según el principio, aplicamos: $u^{i}$ tenemos $\frac{i u^{i}}{u}$
  3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \cos{\left(x \right)}$ :
    1. 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:
    
    $- \frac{i \sin{\left(x \right)} \cos^{i}{\left(x \right)}}{\cos{\left(x \right)}}$
  Como resultado de la secuencia de reglas:
  
  $- \frac{i \sin{\left(x \right)} \cos^{i}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos{\left(x \right)}}$
Como resultado de: $- \frac{i x \sin{\left(x \right)} \cos^{i}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos{\left(x \right)}} + \sin{\left(\cos^{i}{\left(x \right)} \right)}$
Simplificamos:

$- i x \cos^{i}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)} \tan{\left(x \right)} + \sin{\left(\cos^{i}{\left(x \right)} \right)}$

Respuesta:

$- i x \cos^{i}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)} \tan{\left(x \right)} + \sin{\left(\cos^{i}{\left(x \right)} \right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

         I       /   I   \                      
  I*x*cos (x)*cos\cos (x)/*sin(x)      /   I   \
- ------------------------------- + sin\cos (x)/
               cos(x)

$$- \frac{i x \sin{\left(x \right)} \cos^{i}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos{\left(x \right)}} + \sin{\left(\cos^{i}{\left(x \right)} \right)}$$

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

         /                                              /                                                                  2       /   I   \        2       /   I   \        I       2       /   I   \        2*I       2       /   I   \          I       2       /   I   \\                                                                   \
         |                                              |     /   I   \        I       /   I   \          /   I   \   3*sin (x)*cos\cos (x)/   I*sin (x)*cos\cos (x)/   3*cos (x)*sin (x)*sin\cos (x)/   I*cos   (x)*sin (x)*cos\cos (x)/   3*I*cos (x)*sin (x)*sin\cos (x)/|                                                                   |
         |                                            x*|3*cos\cos (x)/ - 3*cos (x)*sin\cos (x)/ + 2*I*cos\cos (x)/ + ---------------------- + ---------------------- - ------------------------------ + -------------------------------- + --------------------------------|*sin(x)                                                            |
         |                        2       /   I   \     |                                                                       2                        2                            2                                 2                                  2                |               I       2       /   I   \          2       /   I   \|
    I    |       /   I   \   3*sin (x)*cos\cos (x)/     \                                                                    cos (x)                  cos (x)                      cos (x)                           cos (x)                            cos (x)             /          3*cos (x)*sin (x)*sin\cos (x)/   3*I*sin (x)*cos\cos (x)/|
-cos (x)*|3*I*cos\cos (x)/ + ---------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------ + ------------------------|
         |                             2                                                                                                                          cos(x)                                                                                                                             2                             2            |
         \                          cos (x)                                                                                                                                                                                                                                                       cos (x)                       cos (x)         /

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

Simplificar

Gráfico

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Expresiones idénticas

Expresiones semejantes

Derivada de x*sin(cos(x)^(i))

Gráfico:

Definida a trozos:

Solución