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y=arctg^2(sinx)

Derivada de y=arctg^2(sinx)

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

Gráfico:

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Solución

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