Derivada de xtg(x)ln(x)

Solución

Ha introducido [src]

x*tan(x)*log(x)

$$x \tan{\left(x \right)} \log{\left(x \right)}$$

(x*tan(x))*log(x)

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = x \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Según el principio, aplicamos: $x$ tenemos $1$
  $g{\left(x \right)} = \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Reescribimos las funciones para diferenciar:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Se aplica la regla de la derivada parcial:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
    1. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
    1. La derivada del coseno es igual a menos el seno:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Ahora aplicamos la regla de la derivada de una divesión:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  Como resultado de: $\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}$
$g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
Como resultado de: $\left(\frac{x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right) \log{\left(x \right)} + \tan{\left(x \right)}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

/  /       2   \         \                
\x*\1 + tan (x)/ + tan(x)/*log(x) + tan(x)

$$\left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \log{\left(x \right)} + \tan{\left(x \right)}$$

Simplificar

Segunda derivada [src]

             /  /       2   \         \                                                  
  tan(x)   2*\x*\1 + tan (x)/ + tan(x)/     /       2        /       2   \       \       
- ------ + ---------------------------- + 2*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/*log(x)
    x                   x

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

Simplificar

Tercera derivada [src]

    /  /       2   \         \                /       2        /       2   \       \                                                        
  3*\x*\1 + tan (x)/ + tan(x)/   2*tan(x)   6*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/     /       2   \ /             /         2   \\       
- ---------------------------- + -------- + ---------------------------------------- + 2*\1 + tan (x)/*\3*tan(x) + x*\1 + 3*tan (x)//*log(x)
                2                    2                         x                                                                            
               x                    x

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

Simplificar

Gráfico

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Derivada de xtg(x)ln(x)

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