Sr Examen

Derivada de cosx/(1+sinx)

Función f() - derivada -er orden en el punto
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Gráfico:

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Definida a trozos:

Solución

Ha introducido [src]
  cos(x)  
----------
1 + sin(x)
cos(x)sin(x)+1\frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1}
cos(x)/(1 + sin(x))
Solución detallada
  1. 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)+1g{\left(x \right)} = \sin{\left(x \right)} + 1.

    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. diferenciamos sin(x)+1\sin{\left(x \right)} + 1 miembro por miembro:

      1. La derivada de una constante 11 es igual a cero.

      2. La derivada del seno es igual al coseno:

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

      Como resultado de: cos(x)\cos{\left(x \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    (sin(x)+1)sin(x)cos2(x)(sin(x)+1)2\frac{- \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)} - \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}

  2. Simplificamos:

    1sin(x)+1- \frac{1}{\sin{\left(x \right)} + 1}


Respuesta:

1sin(x)+1- \frac{1}{\sin{\left(x \right)} + 1}

Gráfica
02468-8-6-4-2-1010-20002000
Primera derivada [src]
                     2      
    sin(x)        cos (x)   
- ---------- - -------------
  1 + sin(x)               2
               (1 + sin(x)) 
sin(x)sin(x)+1cos2(x)(sin(x)+1)2- \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}
Segunda derivada [src]
/          2                          \       
|     2*cos (x)                       |       
|     ---------- + sin(x)             |       
|     1 + sin(x)             2*sin(x) |       
|-1 + ------------------- + ----------|*cos(x)
\          1 + sin(x)       1 + sin(x)/       
----------------------------------------------
                  1 + sin(x)                  
(1+sin(x)+2cos2(x)sin(x)+1sin(x)+1+2sin(x)sin(x)+1)cos(x)sin(x)+1\frac{\left(-1 + \frac{\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 1}}{\sin{\left(x \right)} + 1} + \frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} + 1}\right) \cos{\left(x \right)}}{\sin{\left(x \right)} + 1}
Tercera derivada [src]
                     /                         2     \                                          
                2    |      6*sin(x)      6*cos (x)  |     /     2             \                
             cos (x)*|-1 + ---------- + -------------|     |2*cos (x)          |                
     2               |     1 + sin(x)               2|   3*|---------- + sin(x)|*sin(x)         
3*cos (x)            \                  (1 + sin(x)) /     \1 + sin(x)         /                
---------- - ----------------------------------------- - ------------------------------ + sin(x)
1 + sin(x)                   1 + sin(x)                            1 + sin(x)                   
------------------------------------------------------------------------------------------------
                                           1 + sin(x)                                           
sin(x)3(sin(x)+2cos2(x)sin(x)+1)sin(x)sin(x)+1(1+6sin(x)sin(x)+1+6cos2(x)(sin(x)+1)2)cos2(x)sin(x)+1+3cos2(x)sin(x)+1sin(x)+1\frac{\sin{\left(x \right)} - \frac{3 \left(\sin{\left(x \right)} + \frac{2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sin{\left(x \right)} + 1} - \frac{\left(-1 + \frac{6 \sin{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{6 \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + 1\right)^{2}}\right) \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{3 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + 1}}{\sin{\left(x \right)} + 1}
Gráfico
Derivada de cosx/(1+sinx)