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Derivada de la función/ arctg/ z=arctg((u+v)/(u-v))

Derivada de z=arctg((u+v)/(u-v))

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

Gráfico:

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Solución

Ha introducido [src] 
    /u + v\
atan|-----|
    \u - v/
atan⁡(u+vu−v)\operatorname{atan}{\left(\frac{u + v}{u - v} \right)} 
atan((u + v)/(u - v))
Primera derivada [src] 
  1      u + v  
----- + --------
u - v          2
        (u - v) 
----------------
             2  
      (u + v)   
  1 + --------  
             2  
      (u - v)
1u−v+u+v(u−v)21+(u+v)2(u−v)2\frac{\frac{1}{u - v} + \frac{u + v}{\left(u - v\right)^{2}}}{1 + \frac{\left(u + v\right)^{2}}{\left(u - v\right)^{2}}}
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Segunda derivada [src] 
              /     /    u + v\          \
              |     |1 + -----|*(u + v)  |
  /    u + v\ |     \    u - v/          |
2*|1 + -----|*|1 - ----------------------|
  \    u - v/ |    /           2\        |
              |    |    (u + v) |        |
              |    |1 + --------|*(u - v)|
              |    |           2|        |
              \    \    (u - v) /        /
------------------------------------------
         /           2\                   
         |    (u + v) |        2          
         |1 + --------|*(u - v)           
         |           2|                   
         \    (u - v) /
2(1+u+vu−v)(1−(1+u+vu−v)(u+v)(1+(u+v)2(u−v)2)(u−v))(1+(u+v)2(u−v)2)(u−v)2\frac{2 \left(1 + \frac{u + v}{u - v}\right) \left(1 - \frac{\left(1 + \frac{u + v}{u - v}\right) \left(u + v\right)}{\left(1 + \frac{\left(u + v\right)^{2}}{\left(u - v\right)^{2}}\right) \left(u - v\right)}\right)}{\left(1 + \frac{\left(u + v\right)^{2}}{\left(u - v\right)^{2}}\right) \left(u - v\right)^{2}}
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Tercera derivada [src] 
              /                 2                                                                \
              |        3*(u + v)    4*(u + v)                                         2          |
              |    1 + ---------- + ---------     /    u + v\              /    u + v\         2 |
              |                2      u - v     4*|1 + -----|*(u + v)    4*|1 + -----| *(u + v)  |
  /    u + v\ |         (u - v)                   \    u - v/              \    u - v/           |
2*|1 + -----|*|3 - -------------------------- - ---------------------- + ------------------------|
  \    u - v/ |                      2          /           2\                         2         |
              |               (u + v)           |    (u + v) |           /           2\          |
              |           1 + --------          |1 + --------|*(u - v)   |    (u + v) |         2|
              |                      2          |           2|           |1 + --------| *(u - v) |
              |               (u - v)           \    (u - v) /           |           2|          |
              \                                                          \    (u - v) /          /
--------------------------------------------------------------------------------------------------
                                     /           2\                                               
                                     |    (u + v) |        3                                      
                                     |1 + --------|*(u - v)                                       
                                     |           2|                                               
                                     \    (u - v) /
2(1+u+vu−v)(3−4(1+u+vu−v)(u+v)(1+(u+v)2(u−v)2)(u−v)−1+4(u+v)u−v+3(u+v)2(u−v)21+(u+v)2(u−v)2+4(1+u+vu−v)2(u+v)2(1+(u+v)2(u−v)2)2(u−v)2)(1+(u+v)2(u−v)2)(u−v)3\frac{2 \left(1 + \frac{u + v}{u - v}\right) \left(3 - \frac{4 \left(1 + \frac{u + v}{u - v}\right) \left(u + v\right)}{\left(1 + \frac{\left(u + v\right)^{2}}{\left(u - v\right)^{2}}\right) \left(u - v\right)} - \frac{1 + \frac{4 \left(u + v\right)}{u - v} + \frac{3 \left(u + v\right)^{2}}{\left(u - v\right)^{2}}}{1 + \frac{\left(u + v\right)^{2}}{\left(u - v\right)^{2}}} + \frac{4 \left(1 + \frac{u + v}{u - v}\right)^{2} \left(u + v\right)^{2}}{\left(1 + \frac{\left(u + v\right)^{2}}{\left(u - v\right)^{2}}\right)^{2} \left(u - v\right)^{2}}\right)}{\left(1 + \frac{\left(u + v\right)^{2}}{\left(u - v\right)^{2}}\right) \left(u - v\right)^{3}}
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