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Derivada de la función/ atanx/ y=1÷abtan^-1(b÷atanx)

Derivada de y=1÷abtan^-1(b÷atanx)

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
    /b\     
    |-|     
    \a/     
------------
   /   b   \
tan|-------|
   \atan(x)/
1abtan⁡(batan⁡(x))\frac{\frac{1}{a} b}{\tan{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)}} 
(b/a)/tan(b/atan(x))
Primera derivada [src] 
       2 /       2/   b   \\     
      b *|1 + tan |-------||     
         \        \atan(x)//     
---------------------------------
  /     2\     2       2/   b   \
a*\1 + x /*atan (x)*tan |-------|
                        \atan(x)/
b2(tan⁡2(batan⁡(x))+1)a(x2+1)tan⁡2(batan⁡(x))atan⁡2(x)\frac{b^{2} \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right)}{a \left(x^{2} + 1\right) \tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{2}{\left(x \right)}}
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Segunda derivada [src] 
                          /                                                   /       2/   b   \\ \
                          |                                                 b*|1 + tan |-------|| |
    2 /       2/   b   \\ |   b            x                  1               \        \atan(x)// |
-2*b *|1 + tan |-------||*|-------- + ------------ + -------------------- - ----------------------|
      \        \atan(x)// |    2         /   b   \              /   b   \       2       2/   b   \|
                          |atan (x)   tan|-------|   atan(x)*tan|-------|   atan (x)*tan |-------||
                          \              \atan(x)/              \atan(x)/                \atan(x)//
---------------------------------------------------------------------------------------------------
                                           2                                                       
                                   /     2\      2       /   b   \                                 
                                 a*\1 + x / *atan (x)*tan|-------|                                 
                                                         \atan(x)/
−2b2(tan⁡2(batan⁡(x))+1)(−b(tan⁡2(batan⁡(x))+1)tan⁡2(batan⁡(x))atan⁡2(x)+batan⁡2(x)+xtan⁡(batan⁡(x))+1tan⁡(batan⁡(x))atan⁡(x))a(x2+1)2tan⁡(batan⁡(x))atan⁡2(x)- \frac{2 b^{2} \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right) \left(- \frac{b \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right)}{\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{2}{\left(x \right)}} + \frac{b}{\operatorname{atan}^{2}{\left(x \right)}} + \frac{x}{\tan{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)}} + \frac{1}{\tan{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)}}\right)}{a \left(x^{2} + 1\right)^{2} \tan{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{2}{\left(x \right)}}
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Tercera derivada [src] 
                         /                                                                                                                                                                                                                                                                  2                                                                      \
                         |                                                                                                                                                                           /       2/   b   \\              2 /       2/   b   \\            2 /       2/   b   \\                                                 /       2/   b   \\   |
                         |                            2                                                       2                                                                                  6*b*|1 + tan |-------||           5*b *|1 + tan |-------||         3*b *|1 + tan |-------||                                           6*b*x*|1 + tan |-------||   |
   2 /       2/   b   \\ |        1                2*b                         3                           4*x                          6*b                              6*x                         \        \atan(x)//                \        \atan(x)//              \        \atan(x)//                   6*b*x                         \        \atan(x)//   |
2*b *|1 + tan |-------||*|- ------------- + ----------------- + ------------------------------- + ---------------------- + ------------------------------ + ------------------------------ - ------------------------------- - ------------------------------- + ------------------------------- + ------------------------------ - -------------------------------|
     \        \atan(x)// |     2/   b   \   /     2\     4      /     2\     2       2/   b   \   /     2\    2/   b   \   /     2\     3       /   b   \   /     2\            2/   b   \   /     2\     3       3/   b   \   /     2\     4       2/   b   \   /     2\     4       4/   b   \   /     2\     2       /   b   \   /     2\     2       3/   b   \|
                         |  tan |-------|   \1 + x /*atan (x)   \1 + x /*atan (x)*tan |-------|   \1 + x /*tan |-------|   \1 + x /*atan (x)*tan|-------|   \1 + x /*atan(x)*tan |-------|   \1 + x /*atan (x)*tan |-------|   \1 + x /*atan (x)*tan |-------|   \1 + x /*atan (x)*tan |-------|   \1 + x /*atan (x)*tan|-------|   \1 + x /*atan (x)*tan |-------||
                         \      \atan(x)/                                             \atan(x)/                \atan(x)/                        \atan(x)/                        \atan(x)/                         \atan(x)/                         \atan(x)/                         \atan(x)/                        \atan(x)/                         \atan(x)//
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                                                                                                                                                                          /     2\      2                                                                                                                                                                           
                                                                                                                                                                        a*\1 + x / *atan (x)
2b2(tan⁡2(batan⁡(x))+1)(3b2(tan⁡2(batan⁡(x))+1)2(x2+1)tan⁡4(batan⁡(x))atan⁡4(x)−5b2(tan⁡2(batan⁡(x))+1)(x2+1)tan⁡2(batan⁡(x))atan⁡4(x)+2b2(x2+1)atan⁡4(x)−6bx(tan⁡2(batan⁡(x))+1)(x2+1)tan⁡3(batan⁡(x))atan⁡2(x)+6bx(x2+1)tan⁡(batan⁡(x))atan⁡2(x)−6b(tan⁡2(batan⁡(x))+1)(x2+1)tan⁡3(batan⁡(x))atan⁡3(x)+6b(x2+1)tan⁡(batan⁡(x))atan⁡3(x)+4x2(x2+1)tan⁡2(batan⁡(x))+6x(x2+1)tan⁡2(batan⁡(x))atan⁡(x)−1tan⁡2(batan⁡(x))+3(x2+1)tan⁡2(batan⁡(x))atan⁡2(x))a(x2+1)2atan⁡2(x)\frac{2 b^{2} \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right) \left(\frac{3 b^{2} \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right)^{2}}{\left(x^{2} + 1\right) \tan^{4}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{4}{\left(x \right)}} - \frac{5 b^{2} \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right)}{\left(x^{2} + 1\right) \tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{4}{\left(x \right)}} + \frac{2 b^{2}}{\left(x^{2} + 1\right) \operatorname{atan}^{4}{\left(x \right)}} - \frac{6 b x \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right)}{\left(x^{2} + 1\right) \tan^{3}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{2}{\left(x \right)}} + \frac{6 b x}{\left(x^{2} + 1\right) \tan{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{2}{\left(x \right)}} - \frac{6 b \left(\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} + 1\right)}{\left(x^{2} + 1\right) \tan^{3}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{3}{\left(x \right)}} + \frac{6 b}{\left(x^{2} + 1\right) \tan{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{3}{\left(x \right)}} + \frac{4 x^{2}}{\left(x^{2} + 1\right) \tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)}} + \frac{6 x}{\left(x^{2} + 1\right) \tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}{\left(x \right)}} - \frac{1}{\tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)}} + \frac{3}{\left(x^{2} + 1\right) \tan^{2}{\left(\frac{b}{\operatorname{atan}{\left(x \right)}} \right)} \operatorname{atan}^{2}{\left(x \right)}}\right)}{a \left(x^{2} + 1\right)^{2} \operatorname{atan}^{2}{\left(x \right)}}
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