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y=arctg(cosx)^(1/x)
Derivada de la función/ tg(cosx)/ arctg/ (1/x)/ y=arctg(cosx)^(1/x)

Derivada de y=arctg(cosx)^(1/x)

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
x ______________
\/ atan(cos(x))
atan⁡1x(cos⁡(x))\operatorname{atan}^{\frac{1}{x}}{\left(\cos{\left(x \right)} \right)} 
atan(cos(x))^(1/x)
Solución detallada

  1. No logro encontrar los pasos en la búsqueda de esta derivada.

    Perola derivada

    (log⁡(1x)+1)(1x)1x\left(\log{\left(\frac{1}{x} \right)} + 1\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}

Respuesta:

(log⁡(1x)+1)(1x)1x\left(\log{\left(\frac{1}{x} \right)} + 1\right) \left(\frac{1}{x}\right)^{\frac{1}{x}}

Gráfica
02468-8-6-4-2-1010-500500
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
x ______________ /  log(atan(cos(x)))              sin(x)           \
\/ atan(cos(x)) *|- ----------------- - ----------------------------|
                 |           2            /       2   \             |
                 \          x           x*\1 + cos (x)/*atan(cos(x))/
(−sin⁡(x)x(cos⁡2(x)+1)atan⁡(cos⁡(x))−log⁡(atan⁡(cos⁡(x)))x2)atan⁡1x(cos⁡(x))\left(- \frac{\sin{\left(x \right)}}{x \left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{\log{\left(\operatorname{atan}{\left(\cos{\left(x \right)} \right)} \right)}}{x^{2}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(\cos{\left(x \right)} \right)}
Simplificar
Segunda derivada [src] 
                 /                                                2                                                                                                                                               \
                 |/log(atan(cos(x)))             sin(x)          \                                                                                                                                                |
                 ||----------------- + --------------------------|                                                                                                                                                |
                 ||        x           /       2   \             |                                                                    2                            2                                              |
x ______________ |\                    \1 + cos (x)/*atan(cos(x))/    2*log(atan(cos(x)))             cos(x)                       sin (x)                    2*sin (x)*cos(x)                  2*sin(x)          |
\/ atan(cos(x)) *|------------------------------------------------- + ------------------- - -------------------------- - ---------------------------- - --------------------------- + ----------------------------|
                 |                        x                                     2           /       2   \                             2                              2                  /       2   \             |
                 |                                                             x            \1 + cos (x)/*atan(cos(x))   /       2   \      2           /       2   \                 x*\1 + cos (x)/*atan(cos(x))|
                 \                                                                                                       \1 + cos (x)/ *atan (cos(x))   \1 + cos (x)/ *atan(cos(x))                               /
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                                                                                                         x
(−cos⁡(x)(cos⁡2(x)+1)atan⁡(cos⁡(x))−2sin⁡2(x)cos⁡(x)(cos⁡2(x)+1)2atan⁡(cos⁡(x))−sin⁡2(x)(cos⁡2(x)+1)2atan⁡2(cos⁡(x))+(sin⁡(x)(cos⁡2(x)+1)atan⁡(cos⁡(x))+log⁡(atan⁡(cos⁡(x)))x)2x+2sin⁡(x)x(cos⁡2(x)+1)atan⁡(cos⁡(x))+2log⁡(atan⁡(cos⁡(x)))x2)atan⁡1x(cos⁡(x))x\frac{\left(- \frac{\cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{\sin^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}^{2}{\left(\cos{\left(x \right)} \right)}} + \frac{\left(\frac{\sin{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{\log{\left(\operatorname{atan}{\left(\cos{\left(x \right)} \right)} \right)}}{x}\right)^{2}}{x} + \frac{2 \sin{\left(x \right)}}{x \left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{2 \log{\left(\operatorname{atan}{\left(\cos{\left(x \right)} \right)} \right)}}{x^{2}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(\cos{\left(x \right)} \right)}}{x}
Simplificar
Tercera derivada [src] 
                 /                                                                                                                                                                                                                         /                                                                  2                                                           2               \                                                                                                                                                                                                                                                          \
                 |                                                  3                                                                                                                     /log(atan(cos(x)))             sin(x)          \ |  2*log(atan(cos(x)))             cos(x)                       sin (x)                        2*sin(x)                   2*sin (x)*cos(x)     |                                                                                                                                                                                                                                                          |
                 |  /log(atan(cos(x)))             sin(x)          \                                                                                                                    3*|----------------- + --------------------------|*|- ------------------- + -------------------------- + ---------------------------- - ---------------------------- + ---------------------------|                                                                                                                                                                                                                                                          |
                 |  |----------------- + --------------------------|                                                                                                                      |        x           /       2   \             | |            2           /       2   \                             2                   /       2   \                             2             |                                                                                                                                                                                                                                                          |
                 |  |        x           /       2   \             |                                                                     3                              3                 \                    \1 + cos (x)/*atan(cos(x))/ |           x            \1 + cos (x)/*atan(cos(x))   /       2   \      2           x*\1 + cos (x)/*atan(cos(x))   /       2   \              |             2       3                                                      3                              2                                                                                               2                             2                |
x ______________ |  \                    \1 + cos (x)/*atan(cos(x))/    6*log(atan(cos(x)))             sin(x)                      2*sin (x)                      2*sin (x)                                                               \                                                     \1 + cos (x)/ *atan (cos(x))                                  \1 + cos (x)/ *atan(cos(x))/        8*cos (x)*sin (x)                   6*sin(x)                   6*sin (x)*cos(x)               6*cos (x)*sin(x)              3*cos(x)*sin(x)                    3*cos(x)                       3*sin (x)                     6*sin (x)*cos(x)      |
\/ atan(cos(x)) *|- ------------------------------------------------- - ------------------- + -------------------------- - ---------------------------- + --------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------- - ----------------------------- - ---------------------------- - --------------------------- - ---------------------------- + ---------------------------- + ------------------------------ + -----------------------------|
                 |                           2                                    3           /       2   \                             3                              2                                                                                                                 x                                                                                                                 3                 2 /       2   \                             3                              2                             2                   /       2   \                               2                                2             |
                 |                          x                                    x            \1 + cos (x)/*atan(cos(x))   /       2   \      3           /       2   \                                                                                                                                                                                                                       /       2   \                 x *\1 + cos (x)/*atan(cos(x))   /       2   \      2           /       2   \                 /       2   \      2           x*\1 + cos (x)/*atan(cos(x))     /       2   \      2             /       2   \              |
                 \                                                                                                         \1 + cos (x)/ *atan (cos(x))   \1 + cos (x)/ *atan(cos(x))                                                                                                                                                                                                         \1 + cos (x)/ *atan(cos(x))                                   \1 + cos (x)/ *atan (cos(x))   \1 + cos (x)/ *atan(cos(x))   \1 + cos (x)/ *atan (cos(x))                                  x*\1 + cos (x)/ *atan (cos(x))   x*\1 + cos (x)/ *atan(cos(x))/
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                                                                                                                                                                                                                                                                                                                          x
(sin⁡(x)(cos⁡2(x)+1)atan⁡(cos⁡(x))+2sin⁡3(x)(cos⁡2(x)+1)2atan⁡(cos⁡(x))−6sin⁡(x)cos⁡2(x)(cos⁡2(x)+1)2atan⁡(cos⁡(x))−3sin⁡(x)cos⁡(x)(cos⁡2(x)+1)2atan⁡2(cos⁡(x))−8sin⁡3(x)cos⁡2(x)(cos⁡2(x)+1)3atan⁡(cos⁡(x))−6sin⁡3(x)cos⁡(x)(cos⁡2(x)+1)3atan⁡2(cos⁡(x))−2sin⁡3(x)(cos⁡2(x)+1)3atan⁡3(cos⁡(x))+3(sin⁡(x)(cos⁡2(x)+1)atan⁡(cos⁡(x))+log⁡(atan⁡(cos⁡(x)))x)(cos⁡(x)(cos⁡2(x)+1)atan⁡(cos⁡(x))+2sin⁡2(x)cos⁡(x)(cos⁡2(x)+1)2atan⁡(cos⁡(x))+sin⁡2(x)(cos⁡2(x)+1)2atan⁡2(cos⁡(x))−2sin⁡(x)x(cos⁡2(x)+1)atan⁡(cos⁡(x))−2log⁡(atan⁡(cos⁡(x)))x2)x+3cos⁡(x)x(cos⁡2(x)+1)atan⁡(cos⁡(x))+6sin⁡2(x)cos⁡(x)x(cos⁡2(x)+1)2atan⁡(cos⁡(x))+3sin⁡2(x)x(cos⁡2(x)+1)2atan⁡2(cos⁡(x))−(sin⁡(x)(cos⁡2(x)+1)atan⁡(cos⁡(x))+log⁡(atan⁡(cos⁡(x)))x)3x2−6sin⁡(x)x2(cos⁡2(x)+1)atan⁡(cos⁡(x))−6log⁡(atan⁡(cos⁡(x)))x3)atan⁡1x(cos⁡(x))x\frac{\left(\frac{\sin{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{2 \sin^{3}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{6 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}^{2}{\left(\cos{\left(x \right)} \right)}} - \frac{8 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{3} \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{6 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{3} \operatorname{atan}^{2}{\left(\cos{\left(x \right)} \right)}} - \frac{2 \sin^{3}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{3} \operatorname{atan}^{3}{\left(\cos{\left(x \right)} \right)}} + \frac{3 \left(\frac{\sin{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{\log{\left(\operatorname{atan}{\left(\cos{\left(x \right)} \right)} \right)}}{x}\right) \left(\frac{\cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{\sin^{2}{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}^{2}{\left(\cos{\left(x \right)} \right)}} - \frac{2 \sin{\left(x \right)}}{x \left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{2 \log{\left(\operatorname{atan}{\left(\cos{\left(x \right)} \right)} \right)}}{x^{2}}\right)}{x} + \frac{3 \cos{\left(x \right)}}{x \left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{6 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{x \left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{3 \sin^{2}{\left(x \right)}}{x \left(\cos^{2}{\left(x \right)} + 1\right)^{2} \operatorname{atan}^{2}{\left(\cos{\left(x \right)} \right)}} - \frac{\left(\frac{\sin{\left(x \right)}}{\left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} + \frac{\log{\left(\operatorname{atan}{\left(\cos{\left(x \right)} \right)} \right)}}{x}\right)^{3}}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2} \left(\cos^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\cos{\left(x \right)} \right)}} - \frac{6 \log{\left(\operatorname{atan}{\left(\cos{\left(x \right)} \right)} \right)}}{x^{3}}\right) \operatorname{atan}^{\frac{1}{x}}{\left(\cos{\left(x \right)} \right)}}{x}
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Gráfico
Derivada de y=arctg(cosx)^(1/x)
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