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

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

Función f() - derivada -er orden en el punto
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Solución

Ha introducido [src] 
tan(x)*log(x)
-------------
       x     
      5
log⁡(x)tan⁡(x)5x\frac{\log{\left(x \right)} \tan{\left(x \right)}}{5^{x}} 
(tan(x)*log(x))/5^x
Solución detallada

  1. Se aplica la regla de la derivada parcial:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=log⁡(x)tan⁡(x)f{\left(x \right)} = \log{\left(x \right)} \tan{\left(x \right)} y g(x)=5xg{\left(x \right)} = 5^{x}.

    Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Se aplica la regla de la derivada de una multiplicación:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Derivado log⁡(x)\log{\left(x \right)} es 1x\frac{1}{x}.

      g(x)=tan⁡(x)g{\left(x \right)} = \tan{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Reescribimos las funciones para diferenciar:

        tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

      2. Se aplica la regla de la derivada parcial:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} y g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

        Para calcular ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. La derivada del seno es igual al coseno:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        Para calcular ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. La derivada del coseno es igual a menos el seno:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Ahora aplicamos la regla de la derivada de una divesión:

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

      Como resultado de: (sin⁡2(x)+cos⁡2(x))log⁡(x)cos⁡2(x)+tan⁡(x)x\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 ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. ddx5x=5xlog⁡(5)\frac{d}{d x} 5^{x} = 5^{x} \log{\left(5 \right)}

    Ahora aplicamos la regla de la derivada de una divesión:

    5−2x(5x((sin⁡2(x)+cos⁡2(x))log⁡(x)cos⁡2(x)+tan⁡(x)x)−5xlog⁡(5)log⁡(x)tan⁡(x))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)

  2. Simplificamos:

    5−x(−xlog⁡(5)log⁡(x)tan⁡(x)+xlog⁡(x)cos⁡2(x)+tan⁡(x))x\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:

5−x(−xlog⁡(5)log⁡(x)tan⁡(x)+xlog⁡(x)cos⁡2(x)+tan⁡(x))x\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
02468-8-6-4-2-101020-10
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Primera derivada [src] 
 -x /tan(x)   /       2   \       \    -x                     
5  *|------ + \1 + tan (x)/*log(x)| - 5  *log(5)*log(x)*tan(x)
    \  x                          /
5−x((tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)−5−xlog⁡(5)log⁡(x)tan⁡(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(−2((tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)log⁡(5)+2(tan⁡2(x)+1)log⁡(x)tan⁡(x)+log⁡(5)2log⁡(x)tan⁡(x)+2(tan⁡2(x)+1)x−tan⁡(x)x2)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(3((tan⁡2(x)+1)log⁡(x)+tan⁡(x)x)log⁡(5)2+2(tan⁡2(x)+1)(3tan⁡2(x)+1)log⁡(x)−3(2(tan⁡2(x)+1)log⁡(x)tan⁡(x)+2(tan⁡2(x)+1)x−tan⁡(x)x2)log⁡(5)−log⁡(5)3log⁡(x)tan⁡(x)+6(tan⁡2(x)+1)tan⁡(x)x−3(tan⁡2(x)+1)x2+2tan⁡(x)x3)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
Derivada de y=(tgx*lnx)/5^x
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