Sr Examen

Derivada de y'=ln(cos*sqrt(x))

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

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Definida a trozos:

Solución

Ha introducido [src]
   /         ___\
log\cos(x)*\/ x /
$$\log{\left(\sqrt{x} \cos{\left(x \right)} \right)}$$
log(cos(x)*sqrt(x))
Solución detallada
  1. Sustituimos .

  2. Derivado es .

  3. Luego se aplica una cadena de reglas. Multiplicamos por :

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

      ; calculamos :

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

      ; calculamos :

      1. Según el principio, aplicamos: tenemos

      Como resultado de:

    Como resultado de la secuencia de reglas:

  4. Simplificamos:


Respuesta:

Gráfica
Primera derivada [src]
 cos(x)     ___       
------- - \/ x *sin(x)
    ___               
2*\/ x                
----------------------
       ___            
     \/ x *cos(x)     
$$\frac{- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}}{\sqrt{x} \cos{\left(x \right)}}$$
Segunda derivada [src]
                                     cos(x)       ___          /  cos(x)       ___       \       
                                   - ------ + 2*\/ x *sin(x)   |- ------ + 2*\/ x *sin(x)|*sin(x)
                                       ___                     |    ___                  |       
    ___          sin(x)   cos(x)     \/ x                      \  \/ x                   /       
- \/ x *cos(x) - ------ - ------ + ------------------------- - ----------------------------------
                   ___       3/2              4*x                           2*cos(x)             
                 \/ x     4*x                                                                    
-------------------------------------------------------------------------------------------------
                                             ___                                                 
                                           \/ x *cos(x)                                          
$$\frac{- \sqrt{x} \cos{\left(x \right)} - \frac{\left(2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}}{4 x} - \frac{\sin{\left(x \right)}}{\sqrt{x}} - \frac{\cos{\left(x \right)}}{4 x^{\frac{3}{2}}}}{\sqrt{x} \cos{\left(x \right)}}$$
3-я производная [src]
             /  cos(x)       ___       \   cos(x)       ___          4*sin(x)                            2    /  cos(x)       ___       \   /cos(x)       ___          4*sin(x)\          /  cos(x)       ___       \       
           3*|- ------ + 2*\/ x *sin(x)|   ------ + 4*\/ x *cos(x) + --------                         sin (x)*|- ------ + 2*\/ x *sin(x)|   |------ + 4*\/ x *cos(x) + --------|*sin(x)   |- ------ + 2*\/ x *sin(x)|*sin(x)
             |    ___                  |     3/2                        ___                                   |    ___                  |   |  3/2                        ___  |          |    ___                  |       
  cos(x)     \  \/ x                   /    x                         \/ x      3*sin(x)   3*cos(x)           \  \/ x                   /   \ x                         \/ x   /          \  \/ x                   /       
- ------ - ----------------------------- + ---------------------------------- + -------- + -------- - ----------------------------------- - ------------------------------------------- + ----------------------------------
    ___                    2                              4*x                       3/2        5/2                     2                                      2*cos(x)                                2*x*cos(x)            
  \/ x                  8*x                                                      4*x        8*x                     cos (x)                                                                                                 
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                          ___                                                                                                               
                                                                                                        \/ x *cos(x)                                                                                                        
$$\frac{- \frac{\left(2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(4 \sqrt{x} \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{\sqrt{x}} + \frac{\cos{\left(x \right)}}{x^{\frac{3}{2}}}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{\left(2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}\right) \sin{\left(x \right)}}{2 x \cos{\left(x \right)}} + \frac{4 \sqrt{x} \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{\sqrt{x}} + \frac{\cos{\left(x \right)}}{x^{\frac{3}{2}}}}{4 x} - \frac{3 \left(2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}\right)}{8 x^{2}} - \frac{\cos{\left(x \right)}}{\sqrt{x}} + \frac{3 \sin{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cos{\left(x \right)}}{8 x^{\frac{5}{2}}}}{\sqrt{x} \cos{\left(x \right)}}$$
Tercera derivada [src]
             /  cos(x)       ___       \   cos(x)       ___          4*sin(x)                            2    /  cos(x)       ___       \   /cos(x)       ___          4*sin(x)\          /  cos(x)       ___       \       
           3*|- ------ + 2*\/ x *sin(x)|   ------ + 4*\/ x *cos(x) + --------                         sin (x)*|- ------ + 2*\/ x *sin(x)|   |------ + 4*\/ x *cos(x) + --------|*sin(x)   |- ------ + 2*\/ x *sin(x)|*sin(x)
             |    ___                  |     3/2                        ___                                   |    ___                  |   |  3/2                        ___  |          |    ___                  |       
  cos(x)     \  \/ x                   /    x                         \/ x      3*sin(x)   3*cos(x)           \  \/ x                   /   \ x                         \/ x   /          \  \/ x                   /       
- ------ - ----------------------------- + ---------------------------------- + -------- + -------- - ----------------------------------- - ------------------------------------------- + ----------------------------------
    ___                    2                              4*x                       3/2        5/2                     2                                      2*cos(x)                                2*x*cos(x)            
  \/ x                  8*x                                                      4*x        8*x                     cos (x)                                                                                                 
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                          ___                                                                                                               
                                                                                                        \/ x *cos(x)                                                                                                        
$$\frac{- \frac{\left(2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}\right) \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\left(4 \sqrt{x} \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{\sqrt{x}} + \frac{\cos{\left(x \right)}}{x^{\frac{3}{2}}}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{\left(2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}\right) \sin{\left(x \right)}}{2 x \cos{\left(x \right)}} + \frac{4 \sqrt{x} \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{\sqrt{x}} + \frac{\cos{\left(x \right)}}{x^{\frac{3}{2}}}}{4 x} - \frac{3 \left(2 \sqrt{x} \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{x}}\right)}{8 x^{2}} - \frac{\cos{\left(x \right)}}{\sqrt{x}} + \frac{3 \sin{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cos{\left(x \right)}}{8 x^{\frac{5}{2}}}}{\sqrt{x} \cos{\left(x \right)}}$$
Gráfico
Derivada de y'=ln(cos*sqrt(x))