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Derivada de:
Derivada de (π-2x)³
Derivada de y=e×
Derivada de y*log(5*y-1)
Derivada de y=ln((sqrt(x))+(sqrt(x+a)))
Expresiones idénticas
y=(sinx/cos^2x)^ tres
y es igual a ( seno de x dividir por coseno de al cuadrado x) al cubo
y es igual a ( seno de x dividir por coseno de al cuadrado x) en el grado tres
y=(sinx/cos2x)3
y=sinx/cos2x3
y=(sinx/cos²x)³
y=(sinx/cos en el grado 2x) en el grado 3
y=sinx/cos^2x^3
y=(sinx dividir por cos^2x)^3

Derivada de la función/ cos^2/ y=(sinx/cos^2x)^3

Derivada de y=(sinx/cos^2x)^3

Solución

Ha introducido [src]

         3
/ sin(x)\ 
|-------| 
|   2   | 
\cos (x)/

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

(sin(x)/cos(x)^2)^3

Solución detallada

Sustituimos $u = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$ .
Según el principio, aplicamos: $u^{3}$ tenemos $3 u^{2}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos^{2}{\left(x \right)}$ .
  
  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. 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 $\frac{d}{d x} \cos{\left(x \right)}$ :
    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:
    
    $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos^{3}{\left(x \right)}}{\cos^{4}{\left(x \right)}}$
Como resultado de la secuencia de reglas:

$\frac{3 \left(2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos^{3}{\left(x \right)}\right) \sin^{2}{\left(x \right)}}{\cos^{8}{\left(x \right)}}$
Simplificamos:

$\frac{3 \left(\sin^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{7}{\left(x \right)}}$

Respuesta:

$\frac{3 \left(\sin^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{7}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

           3    /                2   \
   2    sin (x) |3*cos(x)   6*sin (x)|
cos (x)*-------*|-------- + ---------|
           6    |   2           3    |
        cos (x) \cos (x)     cos (x) /
--------------------------------------
                sin(x)

$$\frac{\frac{\sin^{3}{\left(x \right)}}{\cos^{6}{\left(x \right)}} \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{3 \cos{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplificar

Segunda derivada [src]

  /                                              /         2   \             /         2   \\       
  |                                         2    |    6*sin (x)|        2    |    2*sin (x)||       
  |                      2               sin (x)*|5 + ---------|   2*sin (x)*|1 + ---------||       
  |       /         2   \         2              |        2    |             |        2    ||       
  |       |    2*sin (x)|    2*sin (x)           \     cos (x) /             \     cos (x) /|       
3*|-1 + 3*|1 + ---------|  - --------- + ----------------------- - -------------------------|*sin(x)
  |       |        2    |        2                  2                          2            |       
  \       \     cos (x) /     cos (x)            cos (x)                    cos (x)         /       
----------------------------------------------------------------------------------------------------
                                                 4                                                  
                                              cos (x)

$$\frac{3 \left(3 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{2} - \frac{2 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1\right) \sin{\left(x \right)}}{\cos^{4}{\left(x \right)}}$$

Simplificar

Tercera derivada [src]

  /                                                                                                                                                                                                                                    2                                                                                \
  |                                                                                                /         2   \           /          4            2   \              /         2   \             /         2   \     /         2   \                      /         2   \             /         2   \ /         2   \|
  |                                                                                           2    |    6*sin (x)|      2    |    24*sin (x)   28*sin (x)|         4    |    2*sin (x)|        2    |    2*sin (x)|     |    2*sin (x)|     2           4    |    6*sin (x)|        2    |    2*sin (x)| |    6*sin (x)||
  |                      2                                                                 sin (x)*|5 + ---------|   sin (x)*|5 + ---------- + ----------|   10*sin (x)*|1 + ---------|   9*sin (x)*|1 + ---------|   6*|1 + ---------| *sin (x)   2*sin (x)*|5 + ---------|   3*sin (x)*|1 + ---------|*|5 + ---------||
  |       /         2   \         2        /         2   \ /          2            4   \           |        2    |           |        4            2     |              |        2    |             |        2    |     |        2    |                      |        2    |             |        2    | |        2    ||
  |       |    2*sin (x)|    2*sin (x)     |    2*sin (x)| |    13*sin (x)   14*sin (x)|           \     cos (x) /           \     cos (x)      cos (x)  /              \     cos (x) /             \     cos (x) /     \     cos (x) /                      \     cos (x) /             \     cos (x) / \     cos (x) /|
3*|-1 - 3*|1 + ---------|  - --------- + 3*|1 + ---------|*|2 + ---------- + ----------| + ----------------------- + ------------------------------------- - -------------------------- - ------------------------- - -------------------------- + ------------------------- + -----------------------------------------|
  |       |        2    |        2         |        2    | |        2            4     |              2                                2                                 4                            2                           2                            4                                   2                    |
  \       \     cos (x) /     cos (x)      \     cos (x) / \     cos (x)      cos (x)  /           cos (x)                          cos (x)                           cos (x)                      cos (x)                     cos (x)                      cos (x)                             cos (x)                 /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                            3                                                                                                                                                            
                                                                                                                                                         cos (x)

$$\frac{3 \left(- \frac{6 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{2} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 3 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)^{2} + \frac{3 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 3 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \left(\frac{14 \sin^{4}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{13 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) - \frac{10 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin^{4}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - \frac{9 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{4}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(\frac{24 \sin^{4}{\left(x \right)}}{\cos^{4}{\left(x \right)}} + \frac{28 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 1\right)}{\cos^{3}{\left(x \right)}}$$

Simplificar

Gráfico

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Expresiones idénticas

Derivada de y=(sinx/cos^2x)^3

Gráfico:

Definida a trozos:

Solución