Derivada de y=tg³2xcos²2x

Solución

Ha introducido [src]

   32       22   
tan  (x)*cos  (x)

$$\cos^{22}{\left(x \right)} \tan^{32}{\left(x \right)}$$

tan(x)^32*cos(x)^22

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \tan^{32}{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Sustituimos $u = \tan{\left(x \right)}$ .
2. Según el principio, aplicamos: $u^{32}$ tenemos $32 u^{31}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \tan{\left(x \right)}$ :
  1. Reescribimos las funciones para diferenciar:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Se aplica la regla de la derivada parcial:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Como resultado de la secuencia de reglas:
  
  $\frac{32 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{31}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = \cos^{22}{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = \cos{\left(x \right)}$ .
2. Según el principio, aplicamos: $u^{22}$ tenemos $22 u^{21}$
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:
  
  $- 22 \sin{\left(x \right)} \cos^{21}{\left(x \right)}$
Como resultado de: $32 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \cos^{20}{\left(x \right)} \tan^{31}{\left(x \right)} - 22 \sin{\left(x \right)} \cos^{21}{\left(x \right)} \tan^{32}{\left(x \right)}$
Simplificamos:

$\left(32 - 22 \sin^{2}{\left(x \right)}\right) \cos^{20}{\left(x \right)} \tan^{31}{\left(x \right)}$

Respuesta:

$\left(32 - 22 \sin^{2}{\left(x \right)}\right) \cos^{20}{\left(x \right)} \tan^{31}{\left(x \right)}$

Gráfica

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Primera derivada [src]

   22       31    /           2   \         21       32          
cos  (x)*tan  (x)*\32 + 32*tan (x)/ - 22*cos  (x)*tan  (x)*sin(x)

$$\left(32 \tan^{2}{\left(x \right)} + 32\right) \cos^{22}{\left(x \right)} \tan^{31}{\left(x \right)} - 22 \sin{\left(x \right)} \cos^{21}{\left(x \right)} \tan^{32}{\left(x \right)}$$

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Segunda derivada [src]

     20       30    /      2    /     2            2   \         2    /       2   \ /           2   \       /       2   \                     \
2*cos  (x)*tan  (x)*\11*tan (x)*\- cos (x) + 21*sin (x)/ + 16*cos (x)*\1 + tan (x)/*\31 + 33*tan (x)/ - 704*\1 + tan (x)/*cos(x)*sin(x)*tan(x)/

$$2 \left(11 \left(21 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)} + 16 \left(\tan^{2}{\left(x \right)} + 1\right) \left(33 \tan^{2}{\left(x \right)} + 31\right) \cos^{2}{\left(x \right)} - 704 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)}\right) \cos^{20}{\left(x \right)} \tan^{30}{\left(x \right)}$$

Simplificar

Tercera derivada [src]

                    /                                                                           /                             2                           \                                                                                                                        \
     19       29    |        3    /        2             2   \               3    /       2   \ |     4          /       2   \          2    /       2   \|          2    /       2   \ /     2            2   \                 2    /       2   \ /           2   \              |
8*cos  (x)*tan  (x)*\- 11*tan (x)*\- 16*cos (x) + 105*sin (x)/*sin(x) + 8*cos (x)*\1 + tan (x)/*\2*tan (x) + 465*\1 + tan (x)/  + 94*tan (x)*\1 + tan (x)// + 264*tan (x)*\1 + tan (x)/*\- cos (x) + 21*sin (x)/*cos(x) - 264*cos (x)*\1 + tan (x)/*\31 + 33*tan (x)/*sin(x)*tan(x)/

$$8 \left(264 \left(21 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan^{2}{\left(x \right)} - 11 \left(105 \sin^{2}{\left(x \right)} - 16 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \tan^{3}{\left(x \right)} - 264 \left(\tan^{2}{\left(x \right)} + 1\right) \left(33 \tan^{2}{\left(x \right)} + 31\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} \tan{\left(x \right)} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \left(465 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 94 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) \cos^{3}{\left(x \right)}\right) \cos^{19}{\left(x \right)} \tan^{29}{\left(x \right)}$$

Simplificar

Gráfico

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Derivada de y=tg³2xcos²2x

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