Derivada de y=((tg)(ln)√x)

Solución

Ha introducido [src]

                ___
tan(x)*log(x)*\/ x

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

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

Solución detallada

Se aplica la regla de la derivada de una multiplicación:

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

$f{\left(x \right)} = \log{\left(x \right)} \tan{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \tan{\left(x \right)}$ ; 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:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
  1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
  Como resultado de: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x}$
$g{\left(x \right)} = \sqrt{x}$ ; calculamos $\frac{d}{d x} g{\left(x \right)}$ :
1. Según el principio, aplicamos: $\sqrt{x}$ tenemos $\frac{1}{2 \sqrt{x}}$
Como resultado de: $\sqrt{x} \left(\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x}\right) + \frac{\log{\left(x \right)} \tan{\left(x \right)}}{2 \sqrt{x}}$
Simplificamos:

$\frac{x \log{\left(x \right)} + \frac{\log{\left(x \right)} \sin{\left(2 x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{2}}{\sqrt{x} \cos^{2}{\left(x \right)}}$

Respuesta:

$\frac{x \log{\left(x \right)} + \frac{\log{\left(x \right)} \sin{\left(2 x \right)}}{4} + \frac{\sin{\left(2 x \right)}}{2}}{\sqrt{x} \cos^{2}{\left(x \right)}}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

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

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

Simplificar

Segunda derivada [src]

                                                                     tan(x)   /       2   \                       
      /             /       2   \                                \   ------ + \1 + tan (x)/*log(x)                
  ___ |  tan(x)   2*\1 + tan (x)/     /       2   \              |     x                             log(x)*tan(x)
\/ x *|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)| + ----------------------------- - -------------
      |     2            x                                       |                 ___                      3/2   
      \    x                                                     /               \/ x                    4*x

$$\sqrt{x} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right) + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}}{\sqrt{x}} - \frac{\log{\left(x \right)} \tan{\left(x \right)}}{4 x^{\frac{3}{2}}}$$

Simplificar

Tercera derivada [src]

                                                                                                                                               /             /       2   \                                \                  
                                                                                                                                               |  tan(x)   2*\1 + tan (x)/     /       2   \              |                  
                                                                                                           /tan(x)   /       2   \       \   3*|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|                  
      /    /       2   \                                                         /       2   \       \   3*|------ + \1 + tan (x)/*log(x)|     |     2            x                                       |                  
  ___ |  3*\1 + tan (x)/   2*tan(x)     /       2   \ /         2   \          6*\1 + tan (x)/*tan(x)|     \  x                          /     \    x                                                     /   3*log(x)*tan(x)
\/ x *|- --------------- + -------- + 2*\1 + tan (x)/*\1 + 3*tan (x)/*log(x) + ----------------------| - --------------------------------- + -------------------------------------------------------------- + ---------------
      |          2             3                                                         x           |                    3/2                                               ___                                       5/2    
      \         x             x                                                                      /                 4*x                                              2*\/ x                                     8*x

$$\sqrt{x} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{2 \tan{\left(x \right)}}{x^{3}}\right) + \frac{3 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right)}{2 \sqrt{x}} - \frac{3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)}{4 x^{\frac{3}{2}}} + \frac{3 \log{\left(x \right)} \tan{\left(x \right)}}{8 x^{\frac{5}{2}}}$$

Simplificar

Gráfico

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Derivada de y=((tg)(ln)√x)

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