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y'=((ln^6)(tgx))
Derivada de la función/ (tgx)/ y'=((ln^6)(tgx))

Derivada de y'=((ln^6)(tgx))

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

Ha introducido [src] 
   6          
log (x)*tan(x)
log⁡(x)6tan⁡(x)\log{\left(x \right)}^{6} \tan{\left(x \right)} 
log(x)^6*tan(x)
Solución detallada

  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)6f{\left(x \right)} = \log{\left(x \right)}^{6}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Sustituimos u=log⁡(x)u = \log{\left(x \right)}.

    2. Según el principio, aplicamos: u6u^{6} tenemos 6u56 u^{5}

    3. Luego se aplica una cadena de reglas. Multiplicamos por ddxlog⁡(x)\frac{d}{d x} \log{\left(x \right)}:

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

      Como resultado de la secuencia de reglas:

      6log⁡(x)5x\frac{6 \log{\left(x \right)}^{5}}{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)6cos⁡2(x)+6log⁡(x)5tan⁡(x)x\frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(x \right)}^{6}}{\cos^{2}{\left(x \right)}} + \frac{6 \log{\left(x \right)}^{5} \tan{\left(x \right)}}{x}

  2. Simplificamos:

    (xlog⁡(x)+3sin⁡(2x))log⁡(x)5xcos⁡2(x)\frac{\left(x \log{\left(x \right)} + 3 \sin{\left(2 x \right)}\right) \log{\left(x \right)}^{5}}{x \cos^{2}{\left(x \right)}}

Respuesta:

(xlog⁡(x)+3sin⁡(2x))log⁡(x)5xcos⁡2(x)\frac{\left(x \log{\left(x \right)} + 3 \sin{\left(2 x \right)}\right) \log{\left(x \right)}^{5}}{x \cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-2000020000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                             5          
   6    /       2   \   6*log (x)*tan(x)
log (x)*\1 + tan (x)/ + ----------------
                               x
(tan⁡2(x)+1)log⁡(x)6+6log⁡(x)5tan⁡(x)x\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{6} + \frac{6 \log{\left(x \right)}^{5} \tan{\left(x \right)}}{x}
Simplificar
Segunda derivada [src] 
          /                                                          /       2   \       \
     4    |   2    /       2   \          3*(-5 + log(x))*tan(x)   6*\1 + tan (x)/*log(x)|
2*log (x)*|log (x)*\1 + tan (x)/*tan(x) - ---------------------- + ----------------------|
          |                                          2                       x           |
          \                                         x                                    /
2((tan⁡2(x)+1)log⁡(x)2tan⁡(x)+6(tan⁡2(x)+1)log⁡(x)x−3(log⁡(x)−5)tan⁡(x)x2)log⁡(x)42 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{2} \tan{\left(x \right)} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}}{x} - \frac{3 \left(\log{\left(x \right)} - 5\right) \tan{\left(x \right)}}{x^{2}}\right) \log{\left(x \right)}^{4}
Simplificar
Tercera derivada [src] 
          /                                          /                      2   \            /       2   \                              2    /       2   \       \
     3    |   3    /       2   \ /         2   \   3*\20 - 15*log(x) + 2*log (x)/*tan(x)   9*\1 + tan (x)/*(-5 + log(x))*log(x)   18*log (x)*\1 + tan (x)/*tan(x)|
2*log (x)*|log (x)*\1 + tan (x)/*\1 + 3*tan (x)/ + ------------------------------------- - ------------------------------------ + -------------------------------|
          |                                                           3                                      2                                   x               |
          \                                                          x                                      x                                                    /
2((tan⁡2(x)+1)(3tan⁡2(x)+1)log⁡(x)3+18(tan⁡2(x)+1)log⁡(x)2tan⁡(x)x−9(log⁡(x)−5)(tan⁡2(x)+1)log⁡(x)x2+3(2log⁡(x)2−15log⁡(x)+20)tan⁡(x)x3)log⁡(x)32 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{3} + \frac{18 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{2} \tan{\left(x \right)}}{x} - \frac{9 \left(\log{\left(x \right)} - 5\right) \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}}{x^{2}} + \frac{3 \left(2 \log{\left(x \right)}^{2} - 15 \log{\left(x \right)} + 20\right) \tan{\left(x \right)}}{x^{3}}\right) \log{\left(x \right)}^{3}
Simplificar
3-я производная [src] 
          /                                          /                      2   \            /       2   \                              2    /       2   \       \
     3    |   3    /       2   \ /         2   \   3*\20 - 15*log(x) + 2*log (x)/*tan(x)   9*\1 + tan (x)/*(-5 + log(x))*log(x)   18*log (x)*\1 + tan (x)/*tan(x)|
2*log (x)*|log (x)*\1 + tan (x)/*\1 + 3*tan (x)/ + ------------------------------------- - ------------------------------------ + -------------------------------|
          |                                                           3                                      2                                   x               |
          \                                                          x                                      x                                                    /
2((tan⁡2(x)+1)(3tan⁡2(x)+1)log⁡(x)3+18(tan⁡2(x)+1)log⁡(x)2tan⁡(x)x−9(log⁡(x)−5)(tan⁡2(x)+1)log⁡(x)x2+3(2log⁡(x)2−15log⁡(x)+20)tan⁡(x)x3)log⁡(x)32 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{3} + \frac{18 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}^{2} \tan{\left(x \right)}}{x} - \frac{9 \left(\log{\left(x \right)} - 5\right) \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)}}{x^{2}} + \frac{3 \left(2 \log{\left(x \right)}^{2} - 15 \log{\left(x \right)} + 20\right) \tan{\left(x \right)}}{x^{3}}\right) \log{\left(x \right)}^{3}
Simplificar
Gráfico
Derivada de y'=((ln^6)(tgx))
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