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y'=(tsint)^11

Derivada de y'=(tsint)^11

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Gráfico:

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Solución

Ha introducido [src]
          11
(t*sin(t))  
(tsin(t))11\left(t \sin{\left(t \right)}\right)^{11}
(t*sin(t))^11
Solución detallada
  1. Sustituimos u=tsin(t)u = t \sin{\left(t \right)}.

  2. Según el principio, aplicamos: u11u^{11} tenemos 11u1011 u^{10}

  3. Luego se aplica una cadena de reglas. Multiplicamos por ddttsin(t)\frac{d}{d t} t \sin{\left(t \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddtf(t)g(t)=f(t)ddtg(t)+g(t)ddtf(t)\frac{d}{d t} f{\left(t \right)} g{\left(t \right)} = f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}

      f(t)=tf{\left(t \right)} = t; calculamos ddtf(t)\frac{d}{d t} f{\left(t \right)}:

      1. Según el principio, aplicamos: tt tenemos 11

      g(t)=sin(t)g{\left(t \right)} = \sin{\left(t \right)}; calculamos ddtg(t)\frac{d}{d t} g{\left(t \right)}:

      1. La derivada del seno es igual al coseno:

        ddtsin(t)=cos(t)\frac{d}{d t} \sin{\left(t \right)} = \cos{\left(t \right)}

      Como resultado de: tcos(t)+sin(t)t \cos{\left(t \right)} + \sin{\left(t \right)}

    Como resultado de la secuencia de reglas:

    11t10(tcos(t)+sin(t))sin10(t)11 t^{10} \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin^{10}{\left(t \right)}


Respuesta:

11t10(tcos(t)+sin(t))sin10(t)11 t^{10} \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin^{10}{\left(t \right)}

Gráfica
02468-8-6-4-2-1010-5000000000050000000000
Primera derivada [src]
 11    11                             
t  *sin  (t)*(11*sin(t) + 11*t*cos(t))
--------------------------------------
               t*sin(t)               
t11sin11(t)(11tcos(t)+11sin(t))tsin(t)\frac{t^{11} \sin^{11}{\left(t \right)} \left(11 t \cos{\left(t \right)} + 11 \sin{\left(t \right)}\right)}{t \sin{\left(t \right)}}
Segunda derivada [src]
    9    9    /                      2                                                                                              \
11*t *sin (t)*\11*(t*cos(t) + sin(t))  - (t*cos(t) + sin(t))*sin(t) - t*(-2*cos(t) + t*sin(t))*sin(t) - t*(t*cos(t) + sin(t))*cos(t)/
11t9(t(tsin(t)2cos(t))sin(t)t(tcos(t)+sin(t))cos(t)+11(tcos(t)+sin(t))2(tcos(t)+sin(t))sin(t))sin9(t)11 t^{9} \left(- t \left(t \sin{\left(t \right)} - 2 \cos{\left(t \right)}\right) \sin{\left(t \right)} - t \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \cos{\left(t \right)} + 11 \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right)^{2} - \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin{\left(t \right)}\right) \sin^{9}{\left(t \right)}
Tercera derivada [src]
    8    8    /                        2               2                                                 /      2       2    2          2    2                        \    2    2                           2    2                                                    2                 2                                2    2                                                                                                                               2                                     \
11*t *sin (t)*\- 11*(t*cos(t) + sin(t)) *sin(t) - 9*sin (t)*(t*cos(t) + sin(t)) + 11*(t*cos(t) + sin(t))*\10*sin (t) - t *sin (t) + 10*t *cos (t) + 22*t*cos(t)*sin(t)/ + t *sin (t)*(t*cos(t) + sin(t)) - t *sin (t)*(3*sin(t) + t*cos(t)) - 11*t*(t*cos(t) + sin(t)) *cos(t) - 9*t*sin (t)*(-2*cos(t) + t*sin(t)) - 9*t *cos (t)*(t*cos(t) + sin(t)) - 20*t*(t*cos(t) + sin(t))*cos(t)*sin(t) - 11*t*(-2*cos(t) + t*sin(t))*(t*cos(t) + sin(t))*sin(t) - 9*t *(-2*cos(t) + t*sin(t))*cos(t)*sin(t)/
11t8(9t2(tsin(t)2cos(t))sin(t)cos(t)+t2(tcos(t)+sin(t))sin2(t)9t2(tcos(t)+sin(t))cos2(t)t2(tcos(t)+3sin(t))sin2(t)11t(tsin(t)2cos(t))(tcos(t)+sin(t))sin(t)9t(tsin(t)2cos(t))sin2(t)11t(tcos(t)+sin(t))2cos(t)20t(tcos(t)+sin(t))sin(t)cos(t)11(tcos(t)+sin(t))2sin(t)+11(tcos(t)+sin(t))(t2sin2(t)+10t2cos2(t)+22tsin(t)cos(t)+10sin2(t))9(tcos(t)+sin(t))sin2(t))sin8(t)11 t^{8} \left(- 9 t^{2} \left(t \sin{\left(t \right)} - 2 \cos{\left(t \right)}\right) \sin{\left(t \right)} \cos{\left(t \right)} + t^{2} \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin^{2}{\left(t \right)} - 9 t^{2} \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \cos^{2}{\left(t \right)} - t^{2} \left(t \cos{\left(t \right)} + 3 \sin{\left(t \right)}\right) \sin^{2}{\left(t \right)} - 11 t \left(t \sin{\left(t \right)} - 2 \cos{\left(t \right)}\right) \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin{\left(t \right)} - 9 t \left(t \sin{\left(t \right)} - 2 \cos{\left(t \right)}\right) \sin^{2}{\left(t \right)} - 11 t \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right)^{2} \cos{\left(t \right)} - 20 t \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin{\left(t \right)} \cos{\left(t \right)} - 11 \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right)^{2} \sin{\left(t \right)} + 11 \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \left(- t^{2} \sin^{2}{\left(t \right)} + 10 t^{2} \cos^{2}{\left(t \right)} + 22 t \sin{\left(t \right)} \cos{\left(t \right)} + 10 \sin^{2}{\left(t \right)}\right) - 9 \left(t \cos{\left(t \right)} + \sin{\left(t \right)}\right) \sin^{2}{\left(t \right)}\right) \sin^{8}{\left(t \right)}
Gráfico
Derivada de y'=(tsint)^11