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y=(√ uno +sin^2x)x^arctg^3x
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Expresiones semejantes
y=(√1-sin^2x)x^arctg^3x
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Derivada de la función/ arctg/ tg^3x/ sin^2/ y=(√1+sin^2x)x^arctg^3x

Derivada de y=(√1+sin^2x)x^arctg^3x

Solución

Ha introducido [src]

                       3   
/  ___      2   \  atan (x)
\\/ 1  + sin (x)/*x

$$x^{\operatorname{atan}^{3}{\left(x \right)}} \left(\sin^{2}{\left(x \right)} + \sqrt{1}\right)$$

(sqrt(1) + sin(x)^2)*x^(atan(x)^3)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sin^{2}{\left(x \right)} + \sqrt{1}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. diferenciamos $\sin^{2}{\left(x \right)} + \sqrt{1}$ miembro por miembro:
  1. La derivada de una constante $\sqrt{1}$ es igual a cero.
  2. Sustituimos $u = \sin{\left(x \right)}$ .
  3. Según el principio, aplicamos: $u^{2}$ tenemos $2 u$
  4. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de la secuencia de reglas:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  Como resultado de: $2 \sin{\left(x \right)} \cos{\left(x \right)}$
$g{\left(x \right)} = x^{\operatorname{atan}^{3}{\left(x \right)}}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. No logro encontrar los pasos en la búsqueda de esta derivada.
  
  Perola derivada
  
  $\left(\log{\left(\operatorname{atan}^{3}{\left(x \right)} \right)} + 1\right) \left(\operatorname{atan}^{3}{\left(x \right)}\right)^{\operatorname{atan}^{3}{\left(x \right)}}$
Como resultado de: $2 x^{\operatorname{atan}^{3}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} + \left(\log{\left(\operatorname{atan}^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{2}{\left(x \right)} + \sqrt{1}\right) \left(\operatorname{atan}^{3}{\left(x \right)}\right)^{\operatorname{atan}^{3}{\left(x \right)}}$
Simplificamos:

$x^{\operatorname{atan}^{3}{\left(x \right)}} \sin{\left(2 x \right)} + \left(\log{\left(\operatorname{atan}^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{2}{\left(x \right)} + 1\right) \left(\operatorname{atan}^{3}{\left(x \right)}\right)^{\operatorname{atan}^{3}{\left(x \right)}}$

Respuesta:

$x^{\operatorname{atan}^{3}{\left(x \right)}} \sin{\left(2 x \right)} + \left(\log{\left(\operatorname{atan}^{3}{\left(x \right)} \right)} + 1\right) \left(\sin^{2}{\left(x \right)} + 1\right) \left(\operatorname{atan}^{3}{\left(x \right)}\right)^{\operatorname{atan}^{3}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

     3                      /    3            2          \          3                 
 atan (x) /  ___      2   \ |atan (x)   3*atan (x)*log(x)|      atan (x)              
x        *\\/ 1  + sin (x)/*|-------- + -----------------| + 2*x        *cos(x)*sin(x)
                            |   x                  2     |                            
                            \                 1 + x      /

$$x^{\operatorname{atan}^{3}{\left(x \right)}} \left(\frac{3 \log{\left(x \right)} \operatorname{atan}^{2}{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}^{3}{\left(x \right)}}{x}\right) \left(\sin^{2}{\left(x \right)} + \sqrt{1}\right) + 2 x^{\operatorname{atan}^{3}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}$$

Simplificar

Segunda derivada [src]

     3    /                                        /                    2                2                                                 \                                                        \
 atan (x) |       2           2      /       2   \ |/atan(x)   3*log(x)\      3      atan (x)    6*log(x)   6*atan(x)    6*x*atan(x)*log(x)|                 2    /atan(x)   3*log(x)\              |
x        *|- 2*sin (x) + 2*cos (x) + \1 + sin (x)/*||------- + --------| *atan (x) - -------- + --------- + ---------- - ------------------|*atan(x) + 4*atan (x)*|------- + --------|*cos(x)*sin(x)|
          |                                        ||   x            2 |                 2              2     /     2\               2     |                      |   x            2 |              |
          |                                        |\           1 + x  /                x       /     2\    x*\1 + x /       /     2\      |                      \           1 + x  /              |
          \                                        \                                            \1 + x /                     \1 + x /      /                                                        /

$$x^{\operatorname{atan}^{3}{\left(x \right)}} \left(4 \left(\frac{3 \log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right) \sin{\left(x \right)} \cos{\left(x \right)} \operatorname{atan}^{2}{\left(x \right)} + \left(\sin^{2}{\left(x \right)} + 1\right) \left(- \frac{6 x \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\frac{3 \log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)^{2} \operatorname{atan}^{3}{\left(x \right)} + \frac{6 \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{6 \operatorname{atan}{\left(x \right)}}{x \left(x^{2} + 1\right)} - \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\right) \operatorname{atan}{\left(x \right)} - 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)$$

Simplificar

Tercera derivada [src]

     3    /              /                    3                   2            3                         2            2                                             /    2                                                 \                                           2     2          \                                                                             /                    2                2                                                 \                      \
 atan (x) |/       2   \ |/atan(x)   3*log(x)\      6      18*atan (x)   2*atan (x)    6*log(x)    9*atan (x)   6*atan (x)*log(x)         3    /atan(x)   3*log(x)\ |atan (x)    6*log(x)   6*atan(x)    6*x*atan(x)*log(x)|    18*atan(x)   36*x*atan(x)*log(x)   24*x *atan (x)*log(x)|                           2    /   2         2   \ /atan(x)   3*log(x)\     |/atan(x)   3*log(x)\      3      atan (x)    6*log(x)   6*atan(x)    6*x*atan(x)*log(x)|                      |
x        *|\1 + sin (x)/*||------- + --------| *atan (x) - ----------- + ---------- + --------- - ----------- - ----------------- - 3*atan (x)*|------- + --------|*|-------- - --------- - ---------- + ------------------| + ----------- - ------------------- + ---------------------| - 8*cos(x)*sin(x) - 6*atan (x)*\sin (x) - cos (x)/*|------- + --------| + 6*||------- + --------| *atan (x) - -------- + --------- + ---------- - ------------------|*atan(x)*cos(x)*sin(x)|
          |              ||   x            2 |                      2         3               3    2 /     2\               2                  |   x            2 | |    2              2     /     2\               2     |             2                3                      3      |                                                    |   x            2 |     ||   x            2 |                 2              2     /     2\               2     |                      |
          |              |\           1 + x  /              /     2\         x        /     2\    x *\1 + x /       /     2\                   \           1 + x  / |   x       /     2\    x*\1 + x /       /     2\      |     /     2\         /     2\               /     2\       |                                                    \           1 + x  /     |\           1 + x  /                x       /     2\    x*\1 + x /       /     2\      |                      |
          \              \                                  \1 + x /                  \1 + x /                      \1 + x /                                        \           \1 + x /                     \1 + x /      /   x*\1 + x /         \1 + x /               \1 + x /       /                                                                             \                                            \1 + x /                     \1 + x /      /                      /

$$x^{\operatorname{atan}^{3}{\left(x \right)}} \left(- 6 \left(\frac{3 \log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \operatorname{atan}^{2}{\left(x \right)} + \left(\sin^{2}{\left(x \right)} + 1\right) \left(\frac{24 x^{2} \log{\left(x \right)} \operatorname{atan}^{2}{\left(x \right)}}{\left(x^{2} + 1\right)^{3}} - \frac{36 x \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\left(x^{2} + 1\right)^{3}} + \left(\frac{3 \log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)^{3} \operatorname{atan}^{6}{\left(x \right)} - 3 \left(\frac{3 \log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right) \left(\frac{6 x \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{6 \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{6 \operatorname{atan}{\left(x \right)}}{x \left(x^{2} + 1\right)} + \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\right) \operatorname{atan}^{3}{\left(x \right)} - \frac{6 \log{\left(x \right)} \operatorname{atan}^{2}{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{18 \operatorname{atan}^{2}{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{6 \log{\left(x \right)}}{\left(x^{2} + 1\right)^{3}} + \frac{18 \operatorname{atan}{\left(x \right)}}{x \left(x^{2} + 1\right)^{2}} - \frac{9 \operatorname{atan}^{2}{\left(x \right)}}{x^{2} \left(x^{2} + 1\right)} + \frac{2 \operatorname{atan}^{3}{\left(x \right)}}{x^{3}}\right) + 6 \left(- \frac{6 x \log{\left(x \right)} \operatorname{atan}{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\frac{3 \log{\left(x \right)}}{x^{2} + 1} + \frac{\operatorname{atan}{\left(x \right)}}{x}\right)^{2} \operatorname{atan}^{3}{\left(x \right)} + \frac{6 \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \frac{6 \operatorname{atan}{\left(x \right)}}{x \left(x^{2} + 1\right)} - \frac{\operatorname{atan}^{2}{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)} \operatorname{atan}{\left(x \right)} - 8 \sin{\left(x \right)} \cos{\left(x \right)}\right)$$

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Derivada de y=(√1+sin^2x)x^arctg^3x

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