Derivada de y=lncos²5x/sin³5x

Solución

Ha introducido [src]

   25        
log  (cos(x))
-------------
      35     
   sin  (x)

$$\frac{\log{\left(\cos{\left(x \right)} \right)}^{25}}{\sin^{35}{\left(x \right)}}$$

log(cos(x))^25/sin(x)^35

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)}^{25}$ y $g{\left(x \right)} = \sin^{35}{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Sustituimos $u = \log{\left(\cos{\left(x \right)} \right)}$ .
2. Según el principio, aplicamos: $u^{25}$ tenemos $25 u^{24}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \log{\left(\cos{\left(x \right)} \right)}$ :
  1. Sustituimos $u = \cos{\left(x \right)}$ .
  2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
  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:
    
    $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  Como resultado de la secuencia de reglas:
  
  $- \frac{25 \log{\left(\cos{\left(x \right)} \right)}^{24} \sin{\left(x \right)}}{\cos{\left(x \right)}}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = \sin{\left(x \right)}$ .
2. Según el principio, aplicamos: $u^{35}$ tenemos $35 u^{34}$
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Como resultado de la secuencia de reglas:
  
  $35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{- 35 \log{\left(\cos{\left(x \right)} \right)}^{25} \sin^{34}{\left(x \right)} \cos{\left(x \right)} - \frac{25 \log{\left(\cos{\left(x \right)} \right)}^{24} \sin^{36}{\left(x \right)}}{\cos{\left(x \right)}}}{\sin^{70}{\left(x \right)}}$
Simplificamos:

$\frac{\left(- 35 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + 25 \cos{\left(x \right)} - \frac{25}{\cos{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{24}}{\sin^{36}{\left(x \right)}}$

Respuesta:

$\frac{\left(- 35 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + 25 \cos{\left(x \right)} - \frac{25}{\cos{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{24}}{\sin^{36}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

        25                        24               
  35*log  (cos(x))*cos(x)   25*log  (cos(x))*sin(x)
- ----------------------- - -----------------------
             36                           35       
          sin  (x)              cos(x)*sin  (x)

$$- \frac{35 \log{\left(\cos{\left(x \right)} \right)}^{25} \cos{\left(x \right)}}{\sin^{36}{\left(x \right)}} - \frac{25 \log{\left(\cos{\left(x \right)} \right)}^{24} \sin{\left(x \right)}}{\sin^{35}{\left(x \right)} \cos{\left(x \right)}}$$

Simplificar

Segunda derivada [src]

                /                                 /          2   \          2           2               \
     23         |                       2         |    36*cos (x)|   120*sin (x)   5*sin (x)*log(cos(x))|
5*log  (cos(x))*|345*log(cos(x)) + 7*log (cos(x))*|1 + ----------| + ----------- - ---------------------|
                |                                 |        2     |        2                  2          |
                \                                 \     sin (x)  /     cos (x)            cos (x)       /
---------------------------------------------------------------------------------------------------------
                                                    35                                                   
                                                 sin  (x)

$$\frac{5 \left(7 \left(1 + \frac{36 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{2} - \frac{5 \log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 345 \log{\left(\cos{\left(x \right)} \right)} + \frac{120 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{23}}{\sin^{35}{\left(x \right)}}$$

Simplificar

Tercera derivada [src]

                /     /                                       2         2            2            2               \                    /          2   \                  /              2   \              /        2         2                             \                   \
                |     |   2                            276*sin (x)   log (cos(x))*sin (x)   36*sin (x)*log(cos(x))|          2         |    36*cos (x)|        3         |      1332*cos (x)|              |  24*sin (x)   sin (x)*log(cos(x))              |                   |
                |  10*|log (cos(x)) - 36*log(cos(x)) + ----------- + -------------------- - ----------------------|   525*log (cos(x))*|1 + ----------|   7*log (cos(x))*|107 + ------------|*cos(x)   525*|- ---------- + ------------------- + log(cos(x))|*cos(x)*log(cos(x))|
                |     |                                     2                 2                       2           |                    |        2     |                  |           2      |              |      2                 2                       |                   |
     22         |     \                                  cos (x)           cos (x)                 cos (x)        /                    \     sin (x)  /                  \        sin (x)   /              \   cos (x)           cos (x)                    /                   |
5*log  (cos(x))*|- ------------------------------------------------------------------------------------------------ - --------------------------------- - ------------------------------------------ + -------------------------------------------------------------------------|
                |                                               cos(x)                                                              cos(x)                                    2                                                            2                                    |
                \                                                                                                                                                          sin (x)                                                      sin (x)                                 /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                        34                                                                                                                                       
                                                                                                                                     sin  (x)

$$\frac{5 \left(- \frac{525 \left(1 + \frac{36 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{2}}{\cos{\left(x \right)}} - \frac{7 \left(107 + \frac{1332 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{525 \left(\frac{\log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)} - \frac{24 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{10 \left(\frac{\log{\left(\cos{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}^{2} - \frac{36 \log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 36 \log{\left(\cos{\left(x \right)} \right)} + \frac{276 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right)}{\cos{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{22}}{\sin^{34}{\left(x \right)}}$$

Simplificar

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Derivada de y=lncos²5x/sin³5x

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