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y=log(log(logtanx))
Derivada de la función/ y=log/ y=log(log(logtanx))

Derivada de y=log(log(logtanx))

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

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

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

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

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

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

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

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

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

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

      3. Luego se aplica una cadena de reglas. Multiplicamos por ddxtan⁡(x)\frac{d}{d x} \tan{\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 la secuencia de reglas:

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

      Como resultado de la secuencia de reglas:

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

    Como resultado de la secuencia de reglas:

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

  4. Simplificamos:

    2log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))sin⁡(2x)\frac{2}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)} \sin{\left(2 x \right)}}

Respuesta:

2log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))sin⁡(2x)\frac{2}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)} \sin{\left(2 x \right)}}

Gráfica
02468-8-6-4-2-1010-500500
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
                   2               
            1 + tan (x)            
-----------------------------------
log(log(tan(x)))*log(tan(x))*tan(x)
tan⁡2(x)+1log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))tan⁡(x)\frac{\tan^{2}{\left(x \right)} + 1}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}}
Simplificar
Segunda derivada [src] 
              /           2                 2                             2                \
/       2   \ |    1 + tan (x)       1 + tan (x)                   1 + tan (x)             |
\1 + tan (x)/*|2 - ----------- - ------------------- - ------------------------------------|
              |         2                       2                                      2   |
              \      tan (x)     log(tan(x))*tan (x)   log(log(tan(x)))*log(tan(x))*tan (x)/
--------------------------------------------------------------------------------------------
                                log(log(tan(x)))*log(tan(x))
(tan⁡2(x)+1)(−tan⁡2(x)+1tan⁡2(x)−tan⁡2(x)+1log⁡(tan⁡(x))tan⁡2(x)−tan⁡2(x)+1log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))tan⁡2(x)+2)log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - \frac{\tan^{2}{\left(x \right)} + 1}{\log{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)}} - \frac{\tan^{2}{\left(x \right)} + 1}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)} \tan^{2}{\left(x \right)}} + 2\right)}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)}}
Simplificar
Tercera derivada [src] 
              /                                            2                                         2                      2                                                                    2                                       2                                       2          \
              |             /       2   \     /       2   \       /       2   \         /       2   \          /       2   \                 /       2   \                          /       2   \                           /       2   \                           /       2   \           |
/       2   \ |           4*\1 + tan (x)/   2*\1 + tan (x)/     6*\1 + tan (x)/       2*\1 + tan (x)/        3*\1 + tan (x)/               6*\1 + tan (x)/                        2*\1 + tan (x)/                         3*\1 + tan (x)/                         3*\1 + tan (x)/           |
\1 + tan (x)/*|4*tan(x) - --------------- + ---------------- - ------------------ + -------------------- + ------------------- - ----------------------------------- + -------------------------------------- + ------------------------------------ + -------------------------------------|
              |                tan(x)              3           log(tan(x))*tan(x)      2            3                     3      log(log(tan(x)))*log(tan(x))*tan(x)      2                 2            3                                      3                          2            3   |
              \                                 tan (x)                             log (tan(x))*tan (x)   log(tan(x))*tan (x)                                         log (log(tan(x)))*log (tan(x))*tan (x)   log(log(tan(x)))*log(tan(x))*tan (x)   log(log(tan(x)))*log (tan(x))*tan (x)/
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                 log(log(tan(x)))*log(tan(x))
(tan⁡2(x)+1)(2(tan⁡2(x)+1)2tan⁡3(x)+3(tan⁡2(x)+1)2log⁡(tan⁡(x))tan⁡3(x)+2(tan⁡2(x)+1)2log⁡(tan⁡(x))2tan⁡3(x)+3(tan⁡2(x)+1)2log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))tan⁡3(x)+3(tan⁡2(x)+1)2log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))2tan⁡3(x)+2(tan⁡2(x)+1)2log⁡(log⁡(tan⁡(x)))2log⁡(tan⁡(x))2tan⁡3(x)−4(tan⁡2(x)+1)tan⁡(x)−6(tan⁡2(x)+1)log⁡(tan⁡(x))tan⁡(x)−6(tan⁡2(x)+1)log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))tan⁡(x)+4tan⁡(x))log⁡(log⁡(tan⁡(x)))log⁡(tan⁡(x))\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\log{\left(\tan{\left(x \right)} \right)} \tan^{3}{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\log{\left(\tan{\left(x \right)} \right)}^{2} \tan^{3}{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)} \tan^{3}{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)}^{2} \tan^{3}{\left(x \right)}} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)}^{2} \log{\left(\tan{\left(x \right)} \right)}^{2} \tan^{3}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)}{\log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)} \tan{\left(x \right)}} + 4 \tan{\left(x \right)}\right)}{\log{\left(\log{\left(\tan{\left(x \right)} \right)} \right)} \log{\left(\tan{\left(x \right)} \right)}}
Simplificar
Gráfico
Derivada de y=log(log(logtanx))
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