Derivada de y=tanln(x+sinx)

Solución

Ha introducido [src]

tan(x)*log(x + sin(x))

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

tan(x)*log(x + sin(x))

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
$g{\left(x \right)} = \log{\left(x + \sin{\left(x \right)} \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Sustituimos $u = x + \sin{\left(x \right)}$ .
2. Derivado $\log{\left(u \right)}$ es $\frac{1}{u}$ .
3. Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \left(x + \sin{\left(x \right)}\right)$ :
  1. diferenciamos $x + \sin{\left(x \right)}$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. La derivada del seno es igual al coseno:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    Como resultado de: $\cos{\left(x \right)} + 1$
  Como resultado de la secuencia de reglas:
  
  $\frac{\cos{\left(x \right)} + 1}{x + \sin{\left(x \right)}}$
Como resultado de: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x + \sin{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \frac{\left(\cos{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x + \sin{\left(x \right)}}$
Simplificamos:

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Derivada de y=tanln(x+sinx)

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