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y'=(tsint)^11
Derivada de la función/ tsint/ y'=(tsint)^11

Derivada de y'=(tsint)^11

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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))sin⁡10(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))sin⁡10(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
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Primera derivada [src] 
 11    11                             
t  *sin  (t)*(11*sin(t) + 11*t*cos(t))
--------------------------------------
               t*sin(t)
t11sin⁡11(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)}}
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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))sin⁡9(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)}
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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))sin⁡2(t)−9t2(tcos⁡(t)+sin⁡(t))cos⁡2(t)−t2(tcos⁡(t)+3sin⁡(t))sin⁡2(t)−11t(tsin⁡(t)−2cos⁡(t))(tcos⁡(t)+sin⁡(t))sin⁡(t)−9t(tsin⁡(t)−2cos⁡(t))sin⁡2(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))(−t2sin⁡2(t)+10t2cos⁡2(t)+22tsin⁡(t)cos⁡(t)+10sin⁡2(t))−9(tcos⁡(t)+sin⁡(t))sin⁡2(t))sin⁡8(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)}
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Gráfico
Derivada de y'=(tsint)^11
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