Sr Examen

Otras calculadoras

y=lnx/tgx
Derivada de la función/ y=lnx/ x/tgx/ y=lnx/tgx

Derivada de y=lnx/tgx

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
log(x)
------
tan(x)
log⁡(x)tan⁡(x)\frac{\log{\left(x \right)}}{\tan{\left(x \right)}} 
log(x)/tan(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)f{\left(x \right)} = \log{\left(x \right)} y g(x)=tan⁡(x)g{\left(x \right)} = \tan{\left(x \right)}.

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

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

    Para calcular 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)}}

    Ahora aplicamos la regla de la derivada de una divesión:

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

  2. Simplificamos:

    −log⁡(x)sin⁡2(x)+1xtan⁡(x)- \frac{\log{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x \tan{\left(x \right)}}

Respuesta:

−log⁡(x)sin⁡2(x)+1xtan⁡(x)- \frac{\log{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{x \tan{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-2000010000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
           /        2   \       
   1       \-1 - tan (x)/*log(x)
-------- + ---------------------
x*tan(x)             2          
                  tan (x)
(−tan⁡2(x)−1)log⁡(x)tan⁡2(x)+1xtan⁡(x)\frac{\left(- \tan^{2}{\left(x \right)} - 1\right) \log{\left(x \right)}}{\tan^{2}{\left(x \right)}} + \frac{1}{x \tan{\left(x \right)}}
Simplificar
Segunda derivada [src] 
         /       2   \                   /            2   \       
  1    2*\1 + tan (x)/     /       2   \ |     1 + tan (x)|       
- -- - --------------- + 2*\1 + tan (x)/*|-1 + -----------|*log(x)
   2       x*tan(x)                      |          2     |       
  x                                      \       tan (x)  /       
------------------------------------------------------------------
                              tan(x)
2(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)log⁡(x)−2(tan⁡2(x)+1)xtan⁡(x)−1x2tan⁡(x)\frac{2 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x \tan{\left(x \right)}} - \frac{1}{x^{2}}}{\tan{\left(x \right)}}
Simplificar
Tercera derivada [src] 
                                                                                                                 /            2   \
                                                                                                   /       2   \ |     1 + tan (x)|
    /                               2                  3\                                        6*\1 + tan (x)/*|-1 + -----------|
    |                  /       2   \      /       2   \ |                        /       2   \                   |          2     |
    |         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |              2       3*\1 + tan (x)/                   \       tan (x)  /
- 2*|2 + 2*tan (x) - ---------------- + ----------------|*log(x) + --------- + --------------- + ----------------------------------
    |                       2                  4        |           3              2    2                     x*tan(x)             
    \                    tan (x)            tan (x)     /          x *tan(x)      x *tan (x)
−2(3(tan⁡2(x)+1)3tan⁡4(x)−5(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2)log⁡(x)+6(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)xtan⁡(x)+3(tan⁡2(x)+1)x2tan⁡2(x)+2x3tan⁡(x)- 2 \left(\frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{3}}{\tan^{4}{\left(x \right)}} - \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{2}{\left(x \right)}} + 2 \tan^{2}{\left(x \right)} + 2\right) \log{\left(x \right)} + \frac{6 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{x \tan{\left(x \right)}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2} \tan^{2}{\left(x \right)}} + \frac{2}{x^{3} \tan{\left(x \right)}}
Simplificar
Gráfico
Derivada de y=lnx/tgx
    © Sr Examen – Калькулятор