Sr Examen

Otras calculadoras

x/tanx
Derivada de la función/ x/tanx

Derivada de x/tanx

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  x   
------
tan(x)
xtan⁡(x)\frac{x}{\tan{\left(x \right)}} 
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)=xf{\left(x \right)} = x 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. Según el principio, aplicamos: xx tenemos 11

    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:

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

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-5000050000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
           /        2   \
  1      x*\-1 - tan (x)/
------ + ----------------
tan(x)          2        
             tan (x)
x(−tan⁡2(x)−1)tan⁡2(x)+1tan⁡(x)\frac{x \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{1}{\tan{\left(x \right)}}
Simplificar
Segunda derivada [src] 
                /             /            2   \\
  /       2   \ |    1        |     1 + tan (x)||
2*\1 + tan (x)/*|- ------ + x*|-1 + -----------||
                |  tan(x)     |          2     ||
                \             \       tan (x)  //
-------------------------------------------------
                      tan(x)
2(x(tan⁡2(x)+1tan⁡2(x)−1)−1tan⁡(x))(tan⁡2(x)+1)tan⁡(x)\frac{2 \left(x \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) - \frac{1}{\tan{\left(x \right)}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}}
Simplificar
Tercera derivada [src] 
  /                                                                            /            2   \\
  |                                                              /       2   \ |     1 + tan (x)||
  |    /                               2                  3\   3*\1 + tan (x)/*|-1 + -----------||
  |    |                  /       2   \      /       2   \ |                   |          2     ||
  |    |         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |                   \       tan (x)  /|
2*|- x*|2 + 2*tan (x) - ---------------- + ----------------| + ----------------------------------|
  |    |                       2                  4        |                 tan(x)              |
  \    \                    tan (x)            tan (x)     /                                     /
2(−x(3(tan⁡2(x)+1)3tan⁡4(x)−5(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2)+3(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)tan⁡(x))2 \left(- x \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) + \frac{3 \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}}\right)
Simplificar
Gráfico
Derivada de x/tanx
    © Sr Examen – Калькулятор