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y=arctg(4x)/(sinx+2)x=0

Derivada de y=arctg(4x)/(sinx+2)x=0

Función f() - derivada -er orden en el punto
v

Gráfico:

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Solución

Ha introducido [src]
atan(4*x)   
----------*x
sin(x) + 2  
$$x \frac{\operatorname{atan}{\left(4 x \right)}}{\sin{\left(x \right)} + 2}$$
(atan(4*x)/(sin(x) + 2))*x
Gráfica
Primera derivada [src]
  /           4               atan(4*x)*cos(x)\   atan(4*x) 
x*|------------------------ - ----------------| + ----------
  |/        2\                             2  |   sin(x) + 2
  \\1 + 16*x /*(sin(x) + 2)    (sin(x) + 2)   /             
$$x \left(- \frac{\cos{\left(x \right)} \operatorname{atan}{\left(4 x \right)}}{\left(\sin{\left(x \right)} + 2\right)^{2}} + \frac{4}{\left(16 x^{2} + 1\right) \left(\sin{\left(x \right)} + 2\right)}\right) + \frac{\operatorname{atan}{\left(4 x \right)}}{\sin{\left(x \right)} + 2}$$
Segunda derivada [src]
              /               /     2             \                                     \                     
              |               |2*cos (x)          |                                     |                     
              |               |---------- + sin(x)|*atan(4*x)                           |                     
    8         |   128*x       \2 + sin(x)         /                     8*cos(x)        |   2*atan(4*x)*cos(x)
--------- - x*|------------ - ------------------------------- + ------------------------| - ------------------
        2     |           2              2 + sin(x)             /        2\             |       2 + sin(x)    
1 + 16*x      |/        2\                                      \1 + 16*x /*(2 + sin(x))|                     
              \\1 + 16*x /                                                              /                     
--------------------------------------------------------------------------------------------------------------
                                                  2 + sin(x)                                                  
$$\frac{- x \left(\frac{128 x}{\left(16 x^{2} + 1\right)^{2}} - \frac{\left(\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 2}\right) \operatorname{atan}{\left(4 x \right)}}{\sin{\left(x \right)} + 2} + \frac{8 \cos{\left(x \right)}}{\left(16 x^{2} + 1\right) \left(\sin{\left(x \right)} + 2\right)}\right) - \frac{2 \cos{\left(x \right)} \operatorname{atan}{\left(4 x \right)}}{\sin{\left(x \right)} + 2} + \frac{8}{16 x^{2} + 1}}{\sin{\left(x \right)} + 2}$$
Tercera derivada [src]
  /    /           2  \                              /                         2     \                                             \                                                                              
  |    |       64*x   |      /     2             \   |      6*sin(x)      6*cos (x)  |                                             |                                               /     2             \          
  |128*|-1 + ---------|      |2*cos (x)          |   |-1 + ---------- + -------------|*atan(4*x)*cos(x)                            |                                               |2*cos (x)          |          
  |    |             2|   12*|---------- + sin(x)|   |     2 + sin(x)               2|                                             |                                             3*|---------- + sin(x)|*atan(4*x)
  |    \     1 + 16*x /      \2 + sin(x)         /   \                  (2 + sin(x)) /                           384*x*cos(x)      |      384*x              24*cos(x)             \2 + sin(x)         /          
x*|-------------------- + ------------------------ - -------------------------------------------------- + -------------------------| - ------------ - ------------------------ + ---------------------------------
  |               2       /        2\                                    2 + sin(x)                                  2             |              2   /        2\                            2 + sin(x)           
  |    /        2\        \1 + 16*x /*(2 + sin(x))                                                        /        2\              |   /        2\    \1 + 16*x /*(2 + sin(x))                                    
  \    \1 + 16*x /                                                                                        \1 + 16*x / *(2 + sin(x))/   \1 + 16*x /                                                                
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                    2 + sin(x)                                                                                                    
$$\frac{x \left(\frac{384 x \cos{\left(x \right)}}{\left(16 x^{2} + 1\right)^{2} \left(\sin{\left(x \right)} + 2\right)} - \frac{\left(-1 + \frac{6 \sin{\left(x \right)}}{\sin{\left(x \right)} + 2} + \frac{6 \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 2\right)^{2}}\right) \cos{\left(x \right)} \operatorname{atan}{\left(4 x \right)}}{\sin{\left(x \right)} + 2} + \frac{12 \left(\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 2}\right)}{\left(16 x^{2} + 1\right) \left(\sin{\left(x \right)} + 2\right)} + \frac{128 \left(\frac{64 x^{2}}{16 x^{2} + 1} - 1\right)}{\left(16 x^{2} + 1\right)^{2}}\right) - \frac{384 x}{\left(16 x^{2} + 1\right)^{2}} + \frac{3 \left(\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 2}\right) \operatorname{atan}{\left(4 x \right)}}{\sin{\left(x \right)} + 2} - \frac{24 \cos{\left(x \right)}}{\left(16 x^{2} + 1\right) \left(\sin{\left(x \right)} + 2\right)}}{\sin{\left(x \right)} + 2}$$
Gráfico
Derivada de y=arctg(4x)/(sinx+2)x=0