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$y=(sinx)\(1+log(x)/log(10)*sinx)$

¿Cómo usar?
Derivada de:
Derivada de x^-0,2
Derivada de e-x
Derivada de e^e
Derivada de e^((3*x)^2)
Expresiones idénticas
y=(sinx)\(uno +log(x)/log(diez)*sinx)
y es igual a ( seno de x)\(1 más logaritmo de (x) dividir por logaritmo de (10) multiplicar por seno de x)
y es igual a ( seno de x)\(uno más logaritmo de (x) dividir por logaritmo de (diez) multiplicar por seno de x)
y=(sinx)\(1+log(x)/log(10)sinx)
y=sinx\1+logx/log10sinx
y=(sinx)\(1+log(x) dividir por log(10)*sinx)
Expresiones semejantes
y=(sinx)\(1-log(x)/log(10)*sinx)

Derivada de la función/ y=(sinx)\(1+log(x)/log(10)*sinx)

Derivada de y=(sinx)\(1+log(x)/log(10)*sinx)

Solución

Ha introducido [src]

      sin(x)      
------------------
     log(x)       
1 + -------*sin(x)
    log(10)

$$\frac{\sin{\left(x \right)}}{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} \sin{\left(x \right)} + 1}$$

sin(x)/(1 + (log(x)/log(10))*sin(x))

Solución detallada

Se aplica la regla de la derivada parcial:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(10 \right)} \sin{\left(x \right)}$ y $g{\left(x \right)} = \log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
  1. La derivada del seno es igual al coseno:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  Entonces, como resultado: $\log{\left(10 \right)} \cos{\left(x \right)}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. diferenciamos $\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}$ miembro por miembro:
  1. Se aplica la regla de la derivada de una multiplicación:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
    1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
    $g{\left(x \right)} = \sin{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de: $\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}$
  2. La derivada de una constante $\log{\left(10 \right)}$ es igual a cero.
  Como resultado de: $\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}$
Ahora aplicamos la regla de la derivada de una divesión:

$\frac{\left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right) \log{\left(10 \right)} \cos{\left(x \right)} - \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \log{\left(10 \right)} \sin{\left(x \right)}}{\left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right)^{2}}$
Simplificamos:

$\frac{\left(x \log{\left(10 \right)} \cos{\left(x \right)} - \sin^{2}{\left(x \right)}\right) \log{\left(10 \right)}}{x \left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right)^{2}}$

Respuesta:

$\frac{\left(x \log{\left(10 \right)} \cos{\left(x \right)} - \sin^{2}{\left(x \right)}\right) \log{\left(10 \right)}}{x \left(\log{\left(x \right)} \sin{\left(x \right)} + \log{\left(10 \right)}\right)^{2}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                     /    sin(x)    cos(x)*log(x)\       
                     |- --------- - -------------|*sin(x)
      cos(x)         \  x*log(10)      log(10)   /       
------------------ + ------------------------------------
     log(x)                                     2        
1 + -------*sin(x)          /     log(x)       \         
    log(10)                 |1 + -------*sin(x)|         
                            \    log(10)       /

$$\frac{\cos{\left(x \right)}}{\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} \sin{\left(x \right)} + 1} + \frac{\left(- \frac{\log{\left(x \right)} \cos{\left(x \right)}}{\log{\left(10 \right)}} - \frac{\sin{\left(x \right)}}{x \log{\left(10 \right)}}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)}}{\log{\left(10 \right)}} \sin{\left(x \right)} + 1\right)^{2}}$$

Simplificar

Segunda derivada [src]

          /                                                              2\                                           
          |                                      /sin(x)                \ |                                           
          |                                    2*|------ + cos(x)*log(x)| |                                           
          |sin(x)                   2*cos(x)     \  x                   / |                                           
          |------ + log(x)*sin(x) - -------- + ---------------------------|*sin(x)                                    
          |   2                        x       /    log(x)*sin(x)\        |            /sin(x)                \       
          |  x                                 |1 + -------------|*log(10)|          2*|------ + cos(x)*log(x)|*cos(x)
          \                                    \       log(10)   /        /            \  x                   /       
-sin(x) + ------------------------------------------------------------------------ - ---------------------------------
                                /    log(x)*sin(x)\                                     /    log(x)*sin(x)\           
                                |1 + -------------|*log(10)                             |1 + -------------|*log(10)   
                                \       log(10)   /                                     \       log(10)   /           
----------------------------------------------------------------------------------------------------------------------
                                                      log(x)*sin(x)                                                   
                                                  1 + -------------                                                   
                                                         log(10)

$$\frac{- \frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \cos{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \sin{\left(x \right)} + \frac{\left(\frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} + \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}}}{\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1}$$

Simplificar

Tercera derivada [src]

          /                                                                             3      /sin(x)                \ /sin(x)                   2*cos(x)\\                                                                                                                        
          |                                                     /sin(x)                \     6*|------ + cos(x)*log(x)|*|------ + log(x)*sin(x) - --------||                                                /                                                              2\       
          |                                                   6*|------ + cos(x)*log(x)|       \  x                   / |   2                        x    ||                                                |                                      /sin(x)                \ |       
          |                 3*sin(x)   3*cos(x)   2*sin(x)      \  x                   /                                \  x                              /|                                                |                                    2*|------ + cos(x)*log(x)| |       
          |-cos(x)*log(x) - -------- - -------- + -------- + ----------------------------- + --------------------------------------------------------------|*sin(x)                                         |sin(x)                   2*cos(x)     \  x                   / |       
          |                    x           2          3                         2                             /    log(x)*sin(x)\                          |                                              3*|------ + log(x)*sin(x) - -------- + ---------------------------|*cos(x)
          |                               x          x       /    log(x)*sin(x)\     2                        |1 + -------------|*log(10)                  |            /sin(x)                \            |   2                        x       /    log(x)*sin(x)\        |       
          |                                                  |1 + -------------| *log (10)                    \       log(10)   /                          |          3*|------ + cos(x)*log(x)|*sin(x)     |  x                                 |1 + -------------|*log(10)|       
          \                                                  \       log(10)   /                                                                           /            \  x                   /            \                                    \       log(10)   /        /       
-cos(x) - --------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------- + --------------------------------------------------------------------------
                                                                         /    log(x)*sin(x)\                                                                             /    log(x)*sin(x)\                                     /    log(x)*sin(x)\                                
                                                                         |1 + -------------|*log(10)                                                                     |1 + -------------|*log(10)                             |1 + -------------|*log(10)                        
                                                                         \       log(10)   /                                                                             \       log(10)   /                                     \       log(10)   /                                
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                     log(x)*sin(x)                                                                                                                                  
                                                                                                                                 1 + -------------                                                                                                                                  
                                                                                                                                        log(10)

$$\frac{\frac{3 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \cos{\left(x \right)} + \frac{3 \left(\frac{2 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{2}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} + \log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right) \cos{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \frac{\left(\frac{6 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right)^{3}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right)^{2} \log{\left(10 \right)}^{2}} + \frac{6 \left(\log{\left(x \right)} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{x}\right) \left(\log{\left(x \right)} \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x} + \frac{\sin{\left(x \right)}}{x^{2}}\right)}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}} - \log{\left(x \right)} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{x} - \frac{3 \cos{\left(x \right)}}{x^{2}} + \frac{2 \sin{\left(x \right)}}{x^{3}}\right) \sin{\left(x \right)}}{\left(\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1\right) \log{\left(10 \right)}}}{\frac{\log{\left(x \right)} \sin{\left(x \right)}}{\log{\left(10 \right)}} + 1}$$

Simplificar

Gráfico

$Derivada de y=(sinx)\(1+log(x)/log(10)*sinx)$

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Derivada de y=(sinx)\(1+log(x)/log(10)*sinx)

Gráfico:

Definida a trozos:

Solución