Derivada de y=tgx/cosx

Solución

Ha introducido [src]

tan(x)
------
cos(x)

$$\frac{\tan{\left(x \right)}}{\cos{\left(x \right)}}$$

tan(x)/cos(x)

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = \tan{\left(x \right)}$ y $g{\left(x \right)} = \cos{\left(x \right)}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
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:

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

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

Respuesta:

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

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

       2                   
1 + tan (x)   sin(x)*tan(x)
----------- + -------------
   cos(x)           2      
                 cos (x)

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

Simplificar

Segunda derivada [src]

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

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

Simplificar

Tercera derivada [src]

                                                                    /         2   \                                              
                                                                    |    6*sin (x)|                                              
                                                                    |5 + ---------|*sin(x)*tan(x)                                
                                                  /         2   \   |        2    |                   /       2   \              
  /       2   \ /         2   \     /       2   \ |    2*sin (x)|   \     cos (x) /                 6*\1 + tan (x)/*sin(x)*tan(x)
2*\1 + tan (x)/*\1 + 3*tan (x)/ + 3*\1 + tan (x)/*|1 + ---------| + ----------------------------- + -----------------------------
                                                  |        2    |               cos(x)                          cos(x)           
                                                  \     cos (x) /                                                                
---------------------------------------------------------------------------------------------------------------------------------
                                                              cos(x)

$$\frac{3 \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{\left(\frac{6 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 5\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\cos{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)}}{\cos{\left(x \right)}}}{\cos{\left(x \right)}}$$

Simplificar

Gráfico

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Derivada de y=tgx/cosx

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