Sr Examen

Derivada de y=arctgx*ctg3x

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
acot(x)*cot(3*x)
$$\cot{\left(3 x \right)} \operatorname{acot}{\left(x \right)}$$
acot(x)*cot(3*x)
Gráfica
Primera derivada [src]
/          2     \           cot(3*x)
\-3 - 3*cot (3*x)/*acot(x) - --------
                                   2 
                              1 + x  
$$\left(- 3 \cot^{2}{\left(3 x \right)} - 3\right) \operatorname{acot}{\left(x \right)} - \frac{\cot{\left(3 x \right)}}{x^{2} + 1}$$
Segunda derivada [src]
  /  /       2     \                                                  \
  |3*\1 + cot (3*x)/   x*cot(3*x)     /       2     \                 |
2*|----------------- + ---------- + 9*\1 + cot (3*x)/*acot(x)*cot(3*x)|
  |           2                2                                      |
  |      1 + x         /     2\                                       |
  \                    \1 + x /                                       /
$$2 \left(\frac{x \cot{\left(3 x \right)}}{\left(x^{2} + 1\right)^{2}} + 9 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(3 x \right)} \operatorname{acot}{\left(x \right)} + \frac{3 \left(\cot^{2}{\left(3 x \right)} + 1\right)}{x^{2} + 1}\right)$$
Tercera derivada [src]
   //         2 \                                                                                                            \
   ||      4*x  |                                                                                                            |
   ||-1 + ------|*cot(3*x)                                                                                                   |
   ||          2|                /       2     \      /       2     \                                                        |
   |\     1 + x /            9*x*\1 + cot (3*x)/   27*\1 + cot (3*x)/*cot(3*x)      /       2     \ /         2     \        |
-2*|---------------------- + ------------------- + --------------------------- + 27*\1 + cot (3*x)/*\1 + 3*cot (3*x)/*acot(x)|
   |              2                       2                        2                                                         |
   |      /     2\                /     2\                    1 + x                                                          |
   \      \1 + x /                \1 + x /                                                                                   /
$$- 2 \left(\frac{9 x \left(\cot^{2}{\left(3 x \right)} + 1\right)}{\left(x^{2} + 1\right)^{2}} + 27 \left(\cot^{2}{\left(3 x \right)} + 1\right) \left(3 \cot^{2}{\left(3 x \right)} + 1\right) \operatorname{acot}{\left(x \right)} + \frac{27 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(3 x \right)}}{x^{2} + 1} + \frac{\left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \cot{\left(3 x \right)}}{\left(x^{2} + 1\right)^{2}}\right)$$
Gráfico
Derivada de y=arctgx*ctg3x