Sr Examen

Derivada de ctgx*cosx

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

Ha introducido [src]
cot(x)*cos(x)
cos(x)cot(x)\cos{\left(x \right)} \cot{\left(x \right)}
cot(x)*cos(x)
Solución detallada
  1. Se aplica la regla de la derivada de una multiplicación:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=cot(x)f{\left(x \right)} = \cot{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    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. Reescribimos las funciones para diferenciar:

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

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

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

          Ahora aplicamos la regla de la derivada de una divesión:

          sin2(x)+cos2(x)cos2(x)\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        sin2(x)+cos2(x)cos2(x)tan2(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:

        sin2(x)cos2(x)sin2(x)\frac{- \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}

    g(x)=cos(x)g{\left(x \right)} = \cos{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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)}

    Como resultado de: sin2(x)+cos2(x)cos(x)tan2(x)sin(x)cot(x)- \frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos{\left(x \right)} \tan^{2}{\left(x \right)}} - \sin{\left(x \right)} \cot{\left(x \right)}

  2. Simplificamos:

    (1+1sin2(x))cos(x)- \left(1 + \frac{1}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}


Respuesta:

(1+1sin2(x))cos(x)- \left(1 + \frac{1}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}

Gráfica
02468-8-6-4-2-1010-1000010000
Primera derivada [src]
/        2   \                       
\-1 - cot (x)/*cos(x) - cot(x)*sin(x)
(cot2(x)1)cos(x)sin(x)cot(x)\left(- \cot^{2}{\left(x \right)} - 1\right) \cos{\left(x \right)} - \sin{\left(x \right)} \cot{\left(x \right)}
Segunda derivada [src]
                   /       2   \            /       2   \              
-cos(x)*cot(x) + 2*\1 + cot (x)/*sin(x) + 2*\1 + cot (x)/*cos(x)*cot(x)
2(cot2(x)+1)sin(x)+2(cot2(x)+1)cos(x)cot(x)cos(x)cot(x)2 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 2 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \cot{\left(x \right)} - \cos{\left(x \right)} \cot{\left(x \right)}
Tercera derivada [src]
                  /       2   \            /       2   \                   /       2   \ /         2   \       
cot(x)*sin(x) + 3*\1 + cot (x)/*cos(x) - 6*\1 + cot (x)/*cot(x)*sin(x) - 2*\1 + cot (x)/*\1 + 3*cot (x)/*cos(x)
2(cot2(x)+1)(3cot2(x)+1)cos(x)6(cot2(x)+1)sin(x)cot(x)+3(cot2(x)+1)cos(x)+sin(x)cot(x)- 2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - 6 \left(\cot^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \cot{\left(x \right)} + 3 \left(\cot^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} + \sin{\left(x \right)} \cot{\left(x \right)}
Gráfico
Derivada de ctgx*cosx