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Derivada de:
Derivada de x^-7
Derivada de i*n*x
Derivada de 5*tan(x)^3+sin(4*x)*e^((-x)/7)
Derivada de y=-6/x
Expresiones idénticas
y=(tgx*lnx)/ cinco ^x
y es igual a (tgx multiplicar por lnx) dividir por 5 en el grado x
y es igual a (tgx multiplicar por lnx) dividir por cinco en el grado x
y=(tgx*lnx)/5x
y=tgx*lnx/5x
y=(tgxlnx)/5^x
y=(tgxlnx)/5x
y=tgxlnx/5x
y=tgxlnx/5^x
y=(tgx*lnx) dividir por 5^x

Derivada de la función/ x*lnx/ y=(tgx*lnx)/5^x

Derivada de y=(tgx*lnx)/5^x

Solución

Ha introducido [src]

tan(x)*log(x)
-------------
       x     
      5

$$\frac{\log{\left(x \right)} \tan{\left(x \right)}}{5^{x}}$$

(tan(x)*log(x))/5^x

Solución detallada

Se aplica la regla de la derivada parcial:

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

$f{\left(x \right)} = \log{\left(x \right)} \tan{\left(x \right)}$ y $g{\left(x \right)} = 5^{x}$ .

Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
1. Se aplica la regla de la derivada de una multiplicación:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \log{\left(x \right)}$ ; calculamos $\frac{d}{d x} f{\left(x \right)}$ :
  1. Derivado $\log{\left(x \right)}$ es $\frac{1}{x}$ .
  $g{\left(x \right)} = \tan{\left(x \right)}$ ; 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: $\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x}$
Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
1. $\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}$
Ahora aplicamos la regla de la derivada de una divesión:

$5^{- 2 x} \left(5^{x} \left(\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{x}\right) - 5^{x} \log{\left(5 \right)} \log{\left(x \right)} \tan{\left(x \right)}\right)$
Simplificamos:

$\frac{5^{- x} \left(- x \log{\left(5 \right)} \log{\left(x \right)} \tan{\left(x \right)} + \frac{x \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right)}{x}$

Respuesta:

$\frac{5^{- x} \left(- x \log{\left(5 \right)} \log{\left(x \right)} \tan{\left(x \right)} + \frac{x \log{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \tan{\left(x \right)}\right)}{x}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

 -x /tan(x)   /       2   \       \    -x                     
5  *|------ + \1 + tan (x)/*log(x)| - 5  *log(5)*log(x)*tan(x)
    \  x                          /

$$5^{- x} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right) - 5^{- x} \log{\left(5 \right)} \log{\left(x \right)} \tan{\left(x \right)}$$

Simplificar

Segunda derivada [src]

    /                                                        /       2   \                                                        \
 -x |  tan(x)     /tan(x)   /       2   \       \          2*\1 + tan (x)/      2                      /       2   \              |
5  *|- ------ - 2*|------ + \1 + tan (x)/*log(x)|*log(5) + --------------- + log (5)*log(x)*tan(x) + 2*\1 + tan (x)/*log(x)*tan(x)|
    |     2       \  x                          /                 x                                                               |
    \    x                                                                                                                        /

$$5^{- x} \left(- 2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right) \log{\left(5 \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \log{\left(5 \right)}^{2} \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right)$$

Simplificar

Tercera derivada [src]

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

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

Simplificar

Gráfico

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Expresiones idénticas

Derivada de y=(tgx*lnx)/5^x

Gráfico:

Definida a trozos:

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