Sr Examen

Derivada de y=tg(x)/ln(x)/x

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
/tan(x)\
|------|
\log(x)/
--------
   x    
$$\frac{\frac{1}{\log{\left(x \right)}} \tan{\left(x \right)}}{x}$$
(tan(x)/log(x))/x
Solución detallada
  1. Se aplica la regla de la derivada parcial:

    y .

    Para calcular :

    1. Reescribimos las funciones para diferenciar:

    2. Se aplica la regla de la derivada parcial:

      y .

      Para calcular :

      1. La derivada del seno es igual al coseno:

      Para calcular :

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

      Ahora aplicamos la regla de la derivada de una divesión:

    Para calcular :

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

      ; calculamos :

      1. Según el principio, aplicamos: tenemos

      ; calculamos :

      1. Derivado es .

      Como resultado de:

    Ahora aplicamos la regla de la derivada de una divesión:

  2. Simplificamos:


Respuesta:

Gráfica
Primera derivada [src]
       2                           
1 + tan (x)     tan(x)             
----------- - ---------            
   log(x)          2               
              x*log (x)     tan(x) 
----------------------- - ---------
           x               2       
                          x *log(x)
$$\frac{\frac{\tan^{2}{\left(x \right)} + 1}{\log{\left(x \right)}} - \frac{\tan{\left(x \right)}}{x \log{\left(x \right)}^{2}}}{x} - \frac{\tan{\left(x \right)}}{x^{2} \log{\left(x \right)}}$$
Segunda derivada [src]
    /       2       tan(x) \                                                         /      2   \       
  2*|1 + tan (x) - --------|                                         /       2   \   |1 + ------|*tan(x)
    \              x*log(x)/   2*tan(x)     /       2   \          2*\1 + tan (x)/   \    log(x)/       
- -------------------------- + -------- + 2*\1 + tan (x)/*tan(x) - --------------- + -------------------
              x                    2                                   x*log(x)            2            
                                  x                                                       x *log(x)     
--------------------------------------------------------------------------------------------------------
                                                x*log(x)                                                
$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x \log{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1 - \frac{\tan{\left(x \right)}}{x \log{\left(x \right)}}\right)}{x} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \tan{\left(x \right)}}{x^{2} \log{\left(x \right)}} + \frac{2 \tan{\left(x \right)}}{x^{2}}}{x \log{\left(x \right)}}$$
3-я производная [src]
               /                                           /      2   \       \                                                                                                                                                         
               |                           /       2   \   |1 + ------|*tan(x)|                                                                                                                                                         
               |  /       2   \          2*\1 + tan (x)/   \    log(x)/       |                                                                                             /      3         3   \                                      
             3*|2*\1 + tan (x)/*tan(x) - --------------- + -------------------|                                       /       2       tan(x) \                            2*|1 + ------ + -------|*tan(x)     /       2   \ /      2   \
               |                             x*log(x)            2            |                                     6*|1 + tan (x) - --------|     /       2   \            |    log(x)      2   |          3*\1 + tan (x)/*|1 + ------|
  6*tan(x)     \                                                x *log(x)     /     /       2   \ /         2   \     \              x*log(x)/   6*\1 + tan (x)/*tan(x)     \             log (x)/                          \    log(x)/
- -------- - ------------------------------------------------------------------ + 2*\1 + tan (x)/*\1 + 3*tan (x)/ + -------------------------- - ---------------------- - ------------------------------- + ----------------------------
      3                                      x                                                                                   2                      x*log(x)                      3                               2                 
     x                                                                                                                          x                                                    x *log(x)                       x *log(x)          
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                x*log(x)                                                                                                                
$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x \log{\left(x \right)}} - \frac{3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \tan{\left(x \right)}}{x^{2} \log{\left(x \right)}}\right)}{x} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1 - \frac{\tan{\left(x \right)}}{x \log{\left(x \right)}}\right)}{x^{2}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \tan{\left(x \right)}}{x^{3} \log{\left(x \right)}} - \frac{6 \tan{\left(x \right)}}{x^{3}}}{x \log{\left(x \right)}}$$
Tercera derivada [src]
               /                                           /      2   \       \                                                                                                                                                         
               |                           /       2   \   |1 + ------|*tan(x)|                                                                                                                                                         
               |  /       2   \          2*\1 + tan (x)/   \    log(x)/       |                                                                                             /      3         3   \                                      
             3*|2*\1 + tan (x)/*tan(x) - --------------- + -------------------|                                       /       2       tan(x) \                            2*|1 + ------ + -------|*tan(x)     /       2   \ /      2   \
               |                             x*log(x)            2            |                                     6*|1 + tan (x) - --------|     /       2   \            |    log(x)      2   |          3*\1 + tan (x)/*|1 + ------|
  6*tan(x)     \                                                x *log(x)     /     /       2   \ /         2   \     \              x*log(x)/   6*\1 + tan (x)/*tan(x)     \             log (x)/                          \    log(x)/
- -------- - ------------------------------------------------------------------ + 2*\1 + tan (x)/*\1 + 3*tan (x)/ + -------------------------- - ---------------------- - ------------------------------- + ----------------------------
      3                                      x                                                                                   2                      x*log(x)                      3                               2                 
     x                                                                                                                          x                                                    x *log(x)                       x *log(x)          
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                x*log(x)                                                                                                                
$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x \log{\left(x \right)}} - \frac{3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x \log{\left(x \right)}} + \frac{\left(1 + \frac{2}{\log{\left(x \right)}}\right) \tan{\left(x \right)}}{x^{2} \log{\left(x \right)}}\right)}{x} + \frac{3 \left(1 + \frac{2}{\log{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1 - \frac{\tan{\left(x \right)}}{x \log{\left(x \right)}}\right)}{x^{2}} - \frac{2 \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right) \tan{\left(x \right)}}{x^{3} \log{\left(x \right)}} - \frac{6 \tan{\left(x \right)}}{x^{3}}}{x \log{\left(x \right)}}$$
Gráfico
Derivada de y=tg(x)/ln(x)/x