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y=2*x*ln(arctg(4x))

Derivada de y=2*x*ln(arctg(4x))

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

Ha introducido [src]
2*x*log(atan(4*x))
$$2 x \log{\left(\operatorname{atan}{\left(4 x \right)} \right)}$$
(2*x)*log(atan(4*x))
Gráfica
Primera derivada [src]
                            8*x         
2*log(atan(4*x)) + ---------------------
                   /        2\          
                   \1 + 16*x /*atan(4*x)
$$\frac{8 x}{\left(16 x^{2} + 1\right) \operatorname{atan}{\left(4 x \right)}} + 2 \log{\left(\operatorname{atan}{\left(4 x \right)} \right)}$$
Segunda derivada [src]
   /        /    1          \\
   |    2*x*|--------- + 8*x||
   |        \atan(4*x)      /|
16*|1 - ---------------------|
   |                  2      |
   \          1 + 16*x       /
------------------------------
    /        2\               
    \1 + 16*x /*atan(4*x)     
$$\frac{16 \left(- \frac{2 x \left(8 x + \frac{1}{\operatorname{atan}{\left(4 x \right)}}\right)}{16 x^{2} + 1} + 1\right)}{\left(16 x^{2} + 1\right) \operatorname{atan}{\left(4 x \right)}}$$
Tercera derivada [src]
   /                        /                                    2                          \\
   |            3           |               1                64*x               12*x        ||
32*|-24*x - --------- + 8*x*|-1 + ---------------------- + --------- + ---------------------||
   |        atan(4*x)       |     /        2\     2                2   /        2\          ||
   \                        \     \1 + 16*x /*atan (4*x)   1 + 16*x    \1 + 16*x /*atan(4*x)//
----------------------------------------------------------------------------------------------
                                               2                                              
                                    /        2\                                               
                                    \1 + 16*x / *atan(4*x)                                    
$$\frac{32 \left(8 x \left(\frac{64 x^{2}}{16 x^{2} + 1} + \frac{12 x}{\left(16 x^{2} + 1\right) \operatorname{atan}{\left(4 x \right)}} - 1 + \frac{1}{\left(16 x^{2} + 1\right) \operatorname{atan}^{2}{\left(4 x \right)}}\right) - 24 x - \frac{3}{\operatorname{atan}{\left(4 x \right)}}\right)}{\left(16 x^{2} + 1\right)^{2} \operatorname{atan}{\left(4 x \right)}}$$
Gráfico
Derivada de y=2*x*ln(arctg(4x))