Sr Examen

Otras calculadoras

y=tan(cos(5ctgx))
Derivada de la función/ y=tan/ 5ctgx/ y=tan(cos(5ctgx))

Derivada de y=tan(cos(5ctgx))

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
tan(cos(5*cot(x)))
tan⁡(cos⁡(5cot⁡(x)))\tan{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} 
tan(cos(5*cot(x)))
Solución detallada

  1. Reescribimos las funciones para diferenciar:

    tan⁡(cos⁡(5cot⁡(x)))=sin⁡(cos⁡(5cot⁡(x)))cos⁡(cos⁡(5cot⁡(x)))\tan{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} = \frac{\sin{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)}}{\cos{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)}}

  2. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

    f(x)=sin⁡(cos⁡(5cot⁡(x)))f{\left(x \right)} = \sin{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} y g(x)=cos⁡(cos⁡(5cot⁡(x)))g{\left(x \right)} = \cos{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Sustituimos u=cos⁡(5cot⁡(x))u = \cos{\left(5 \cot{\left(x \right)} \right)}.

    2. La derivada del seno es igual al coseno:

      ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxcos⁡(5cot⁡(x))\frac{d}{d x} \cos{\left(5 \cot{\left(x \right)} \right)}:

      1. Sustituimos u=5cot⁡(x)u = 5 \cot{\left(x \right)}.

      2. La derivada del coseno es igual a menos el seno:

        dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddx5cot⁡(x)\frac{d}{d x} 5 \cot{\left(x \right)}:

        1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

          1. Hay varias formas de calcular esta derivada.

            Method #1

            1. Reescribimos las funciones para diferenciar:

              cot⁡(x)=1tan⁡(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

            2. Sustituimos u=tan⁡(x)u = \tan{\left(x \right)}.

            3. Según el principio, aplicamos: 1u\frac{1}{u} tenemos −1u2- \frac{1}{u^{2}}

            4. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

              1. ddxtan⁡(x)=1cos⁡2(x)\frac{d}{d x} \tan{\left(x \right)} = \frac{1}{\cos^{2}{\left(x \right)}}

              Como resultado de la secuencia de reglas:

              −sin⁡2(x)+cos⁡2(x)cos⁡2(x)tan⁡2(x)- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

            Method #2

            1. Reescribimos las funciones para diferenciar:

              cot⁡(x)=cos⁡(x)sin⁡(x)\cot{\left(x \right)} = \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

            2. Se aplica la regla de la derivada parcial:

              ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}

              f(x)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)} y g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

              Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

              1. La derivada del coseno es igual a menos el seno:

                ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

              Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

              1. La derivada del seno es igual al coseno:

                ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

              Ahora aplicamos la regla de la derivada de una divesión:

              −sin⁡2(x)−cos⁡2(x)sin⁡2(x)\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}

          Entonces, como resultado: −5(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        5(sin⁡2(x)+cos⁡2(x))sin⁡(5cot⁡(x))cos⁡2(x)tan⁡2(x)\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(5 \cot{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

      Como resultado de la secuencia de reglas:

      5(sin⁡2(x)+cos⁡2(x))sin⁡(5cot⁡(x))cos⁡(cos⁡(5cot⁡(x)))cos⁡2(x)tan⁡2(x)\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(5 \cot{\left(x \right)} \right)} \cos{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

    Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Sustituimos u=cos⁡(5cot⁡(x))u = \cos{\left(5 \cot{\left(x \right)} \right)}.

    2. La derivada del coseno es igual a menos el seno:

      dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxcos⁡(5cot⁡(x))\frac{d}{d x} \cos{\left(5 \cot{\left(x \right)} \right)}:

      1. Sustituimos u=5cot⁡(x)u = 5 \cot{\left(x \right)}.

      2. La derivada del coseno es igual a menos el seno:

        dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddx5cot⁡(x)\frac{d}{d x} 5 \cot{\left(x \right)}:

        1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

          1. ddxcot⁡(x)=−1sin⁡2(x)\frac{d}{d x} \cot{\left(x \right)} = - \frac{1}{\sin^{2}{\left(x \right)}}

          Entonces, como resultado: −5(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        5(sin⁡2(x)+cos⁡2(x))sin⁡(5cot⁡(x))cos⁡2(x)tan⁡2(x)\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(5 \cot{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

      Como resultado de la secuencia de reglas:

      −5(sin⁡2(x)+cos⁡2(x))sin⁡(cos⁡(5cot⁡(x)))sin⁡(5cot⁡(x))cos⁡2(x)tan⁡2(x)- \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} \sin{\left(5 \cot{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

    Ahora aplicamos la regla de la derivada de una divesión:

    5(sin⁡2(x)+cos⁡2(x))sin⁡2(cos⁡(5cot⁡(x)))sin⁡(5cot⁡(x))cos⁡2(x)tan⁡2(x)+5(sin⁡2(x)+cos⁡2(x))sin⁡(5cot⁡(x))cos⁡2(cos⁡(5cot⁡(x)))cos⁡2(x)tan⁡2(x)cos⁡2(cos⁡(5cot⁡(x)))\frac{\frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} \sin{\left(5 \cot{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}} + \frac{5 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(5 \cot{\left(x \right)} \right)} \cos^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)}}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}}{\cos^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)}}

  3. Simplificamos:

    5sin⁡(5tan⁡(x))sin⁡2(x)cos⁡2(cos⁡(5tan⁡(x)))\frac{5 \sin{\left(\frac{5}{\tan{\left(x \right)}} \right)}}{\sin^{2}{\left(x \right)} \cos^{2}{\left(\cos{\left(\frac{5}{\tan{\left(x \right)}} \right)} \right)}}

Respuesta:

5sin⁡(5tan⁡(x))sin⁡2(x)cos⁡2(cos⁡(5tan⁡(x)))\frac{5 \sin{\left(\frac{5}{\tan{\left(x \right)}} \right)}}{\sin^{2}{\left(x \right)} \cos^{2}{\left(\cos{\left(\frac{5}{\tan{\left(x \right)}} \right)} \right)}}

Gráfica
02468-8-6-4-2-1010-100000100000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
 /       2               \ /          2   \              
-\1 + tan (cos(5*cot(x)))/*\-5 - 5*cot (x)/*sin(5*cot(x))
−(tan⁡2(cos⁡(5cot⁡(x)))+1)(−5cot⁡2(x)−5)sin⁡(5cot⁡(x))- \left(\tan^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} + 1\right) \left(- 5 \cot^{2}{\left(x \right)} - 5\right) \sin{\left(5 \cot{\left(x \right)} \right)}
Simplificar
Segunda derivada [src] 
  /       2   \ /       2               \ /    /       2   \                                                2           /       2   \                   \
5*\1 + cot (x)/*\1 + tan (cos(5*cot(x)))/*\- 5*\1 + cot (x)/*cos(5*cot(x)) - 2*cot(x)*sin(5*cot(x)) + 10*sin (5*cot(x))*\1 + cot (x)/*tan(cos(5*cot(x)))/
5(tan⁡2(cos⁡(5cot⁡(x)))+1)(cot⁡2(x)+1)(10(cot⁡2(x)+1)sin⁡2(5cot⁡(x))tan⁡(cos⁡(5cot⁡(x)))−5(cot⁡2(x)+1)cos⁡(5cot⁡(x))−2sin⁡(5cot⁡(x))cot⁡(x))5 \left(\tan^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \left(10 \left(\cot^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(5 \cot{\left(x \right)} \right)} \tan{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} - 5 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(5 \cot{\left(x \right)} \right)} - 2 \sin{\left(5 \cot{\left(x \right)} \right)} \cot{\left(x \right)}\right)
Simplificar
Tercera derivada [src] 
                                          /                  2                                                                                                                                   2                                                             2                                                       2                                                                                                           \
  /       2   \ /       2               \ |     /       2   \                    /       2   \                      2                       /       2   \                           /       2   \     3           /       2               \       /       2   \     3              2                      /       2   \                                                         2           /       2   \                          |
5*\1 + cot (x)/*\1 + tan (cos(5*cot(x)))/*\- 25*\1 + cot (x)/ *sin(5*cot(x)) + 2*\1 + cot (x)/*sin(5*cot(x)) + 4*cot (x)*sin(5*cot(x)) + 30*\1 + cot (x)/*cos(5*cot(x))*cot(x) + 50*\1 + cot (x)/ *sin (5*cot(x))*\1 + tan (cos(5*cot(x)))/ + 100*\1 + cot (x)/ *sin (5*cot(x))*tan (cos(5*cot(x))) - 150*\1 + cot (x)/ *cos(5*cot(x))*sin(5*cot(x))*tan(cos(5*cot(x))) - 60*sin (5*cot(x))*\1 + cot (x)/*cot(x)*tan(cos(5*cot(x)))/
5(tan⁡2(cos⁡(5cot⁡(x)))+1)(cot⁡2(x)+1)(50(tan⁡2(cos⁡(5cot⁡(x)))+1)(cot⁡2(x)+1)2sin⁡3(5cot⁡(x))+100(cot⁡2(x)+1)2sin⁡3(5cot⁡(x))tan⁡2(cos⁡(5cot⁡(x)))−150(cot⁡2(x)+1)2sin⁡(5cot⁡(x))cos⁡(5cot⁡(x))tan⁡(cos⁡(5cot⁡(x)))−25(cot⁡2(x)+1)2sin⁡(5cot⁡(x))−60(cot⁡2(x)+1)sin⁡2(5cot⁡(x))tan⁡(cos⁡(5cot⁡(x)))cot⁡(x)+2(cot⁡2(x)+1)sin⁡(5cot⁡(x))+30(cot⁡2(x)+1)cos⁡(5cot⁡(x))cot⁡(x)+4sin⁡(5cot⁡(x))cot⁡2(x))5 \left(\tan^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \left(50 \left(\tan^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \sin^{3}{\left(5 \cot{\left(x \right)} \right)} + 100 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \sin^{3}{\left(5 \cot{\left(x \right)} \right)} \tan^{2}{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} - 150 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(5 \cot{\left(x \right)} \right)} \cos{\left(5 \cot{\left(x \right)} \right)} \tan{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} - 25 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \sin{\left(5 \cot{\left(x \right)} \right)} - 60 \left(\cot^{2}{\left(x \right)} + 1\right) \sin^{2}{\left(5 \cot{\left(x \right)} \right)} \tan{\left(\cos{\left(5 \cot{\left(x \right)} \right)} \right)} \cot{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(5 \cot{\left(x \right)} \right)} + 30 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(5 \cot{\left(x \right)} \right)} \cot{\left(x \right)} + 4 \sin{\left(5 \cot{\left(x \right)} \right)} \cot^{2}{\left(x \right)}\right)
Simplificar
Gráfico
Derivada de y=tan(cos(5ctgx))
    © Sr Examen – Калькулятор