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  • Derivada de 4*y Derivada de 4*y
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  • cuatro ^tan(x)^(tres)*acos(tres)/x
  • 4 en el grado tangente de (x) en el grado (3) multiplicar por arco coseno de eno de (3) dividir por x
  • cuatro en el grado tangente de (x) en el grado (tres) multiplicar por arco coseno de eno de (tres) dividir por x
  • 4tan(x)(3)*acos(3)/x
  • 4tanx3*acos3/x
  • 4^tan(x)^(3)acos(3)/x
  • 4tan(x)(3)acos(3)/x
  • 4tanx3acos3/x
  • 4^tanx^3acos3/x
  • 4^tan(x)^(3)*acos(3) dividir por x

  • Expresiones semejantes

  • 4^tan(x)^(3)*arccos(3)/x
Derivada de la función/ tan(x)/ 4^tan(x)^(3)*acos(3)/x

Derivada de 4^tan(x)^(3)*acos(3)/x

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

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
    3           
 tan (x)        
4       *acos(3)
----------------
       x
4tan⁡3(x)acos⁡(3)x\frac{4^{\tan^{3}{\left(x \right)}} \operatorname{acos}{\left(3 \right)}}{x} 
(4^(tan(x)^3)*acos(3))/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)=4tan⁡3(x)acos⁡(3)f{\left(x \right)} = 4^{\tan^{3}{\left(x \right)}} \operatorname{acos}{\left(3 \right)} y g(x)=xg{\left(x \right)} = x.

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

    1. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.

      1. Sustituimos u=tan⁡3(x)u = \tan^{3}{\left(x \right)}.

      2. ddu4u=4ulog⁡(4)\frac{d}{d u} 4^{u} = 4^{u} \log{\left(4 \right)}

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

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

        2. Según el principio, aplicamos: u3u^{3} tenemos 3u23 u^{2}

        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:

          3(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡2(x)\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

        Como resultado de la secuencia de reglas:

        3⋅4tan⁡3(x)(sin⁡2(x)+cos⁡2(x))log⁡(4)tan⁡2(x)cos⁡2(x)\frac{3 \cdot 4^{\tan^{3}{\left(x \right)}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(4 \right)} \tan^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}

      Entonces, como resultado: 3⋅4tan⁡3(x)(sin⁡2(x)+cos⁡2(x))log⁡(4)tan⁡2(x)acos⁡(3)cos⁡2(x)\frac{3 \cdot 4^{\tan^{3}{\left(x \right)}} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(4 \right)} \tan^{2}{\left(x \right)} \operatorname{acos}{\left(3 \right)}}{\cos^{2}{\left(x \right)}}

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

    1. Según el principio, aplicamos: xx tenemos 11

    Ahora aplicamos la regla de la derivada de una divesión:

    3⋅4tan⁡3(x)x(sin⁡2(x)+cos⁡2(x))log⁡(4)tan⁡2(x)acos⁡(3)cos⁡2(x)−4tan⁡3(x)acos⁡(3)x2\frac{\frac{3 \cdot 4^{\tan^{3}{\left(x \right)}} x \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \log{\left(4 \right)} \tan^{2}{\left(x \right)} \operatorname{acos}{\left(3 \right)}}{\cos^{2}{\left(x \right)}} - 4^{\tan^{3}{\left(x \right)}} \operatorname{acos}{\left(3 \right)}}{x^{2}}

  2. Simplificamos:

    4tan⁡3(x)(6xlog⁡(2)sin⁡2(x)cos⁡4(x)−1)acos⁡(3)x2\frac{4^{\tan^{3}{\left(x \right)}} \left(\frac{6 x \log{\left(2 \right)} \sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - 1\right) \operatorname{acos}{\left(3 \right)}}{x^{2}}

Respuesta:

4tan⁡3(x)(6xlog⁡(2)sin⁡2(x)cos⁡4(x)−1)acos⁡(3)x2\frac{4^{\tan^{3}{\left(x \right)}} \left(\frac{6 x \log{\left(2 \right)} \sin^{2}{\left(x \right)}}{\cos^{4}{\left(x \right)}} - 1\right) \operatorname{acos}{\left(3 \right)}}{x^{2}}

Primera derivada [src] 
      3                  3                                          
   tan (x)            tan (x)    2    /         2   \               
  4       *acos(3)   4       *tan (x)*\3 + 3*tan (x)/*acos(3)*log(4)
- ---------------- + -----------------------------------------------
          2                                 x                       
         x
4tan⁡3(x)(3tan⁡2(x)+3)log⁡(4)tan⁡2(x)acos⁡(3)x−4tan⁡3(x)acos⁡(3)x2\frac{4^{\tan^{3}{\left(x \right)}} \left(3 \tan^{2}{\left(x \right)} + 3\right) \log{\left(4 \right)} \tan^{2}{\left(x \right)} \operatorname{acos}{\left(3 \right)}}{x} - \frac{4^{\tan^{3}{\left(x \right)}} \operatorname{acos}{\left(3 \right)}}{x^{2}}
Simplificar
Segunda derivada [src] 
    3    /          2    /       2   \                                                                                        \        
 tan (x) |2    6*tan (x)*\1 + tan (x)/*log(4)     /       2   \ /         2           3    /       2   \       \              |        
4       *|-- - ------------------------------ + 3*\1 + tan (x)/*\2 + 4*tan (x) + 3*tan (x)*\1 + tan (x)/*log(4)/*log(4)*tan(x)|*acos(3)
         | 2                 x                                                                                                |        
         \x                                                                                                                   /        
---------------------------------------------------------------------------------------------------------------------------------------
                                                                   x
4tan⁡3(x)(3(tan⁡2(x)+1)(3(tan⁡2(x)+1)log⁡(4)tan⁡3(x)+4tan⁡2(x)+2)log⁡(4)tan⁡(x)−6(tan⁡2(x)+1)log⁡(4)tan⁡2(x)x+2x2)acos⁡(3)x\frac{4^{\tan^{3}{\left(x \right)}} \left(3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(4 \right)} \tan^{3}{\left(x \right)} + 4 \tan^{2}{\left(x \right)} + 2\right) \log{\left(4 \right)} \tan{\left(x \right)} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(4 \right)} \tan^{2}{\left(x \right)}}{x} + \frac{2}{x^{2}}\right) \operatorname{acos}{\left(3 \right)}}{x}
Simplificar
Tercera derivada [src] 
      3    /                     /               2                                                         2                                   2                                                 \               2    /       2   \            /       2   \ /         2           3    /       2   \       \              \        
   tan (x) |  2    /       2   \ |  /       2   \         4            2    /       2   \     /       2   \     2       6         /       2   \     3                   5    /       2   \       |          6*tan (x)*\1 + tan (x)/*log(4)   3*\1 + tan (x)/*\2 + 4*tan (x) + 3*tan (x)*\1 + tan (x)/*log(4)/*log(4)*tan(x)|        
3*4       *|- -- + \1 + tan (x)/*\2*\1 + tan (x)/  + 4*tan (x) + 14*tan (x)*\1 + tan (x)/ + 9*\1 + tan (x)/ *log (4)*tan (x) + 18*\1 + tan (x)/ *tan (x)*log(4) + 18*tan (x)*\1 + tan (x)/*log(4)/*log(4) + ------------------------------ - ------------------------------------------------------------------------------|*acos(3)
           |   3                                                                                                                                                                                                           2                                                       x                                       |        
           \  x                                                                                                                                                                                                           x                                                                                                /        
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                                 x
3⋅4tan⁡3(x)((tan⁡2(x)+1)(9(tan⁡2(x)+1)2log⁡(4)2tan⁡6(x)+18(tan⁡2(x)+1)2log⁡(4)tan⁡3(x)+2(tan⁡2(x)+1)2+18(tan⁡2(x)+1)log⁡(4)tan⁡5(x)+14(tan⁡2(x)+1)tan⁡2(x)+4tan⁡4(x))log⁡(4)−3(tan⁡2(x)+1)(3(tan⁡2(x)+1)log⁡(4)tan⁡3(x)+4tan⁡2(x)+2)log⁡(4)tan⁡(x)x+6(tan⁡2(x)+1)log⁡(4)tan⁡2(x)x2−2x3)acos⁡(3)x\frac{3 \cdot 4^{\tan^{3}{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \left(9 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(4 \right)}^{2} \tan^{6}{\left(x \right)} + 18 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(4 \right)} \tan^{3}{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 18 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(4 \right)} \tan^{5}{\left(x \right)} + 14 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \tan^{4}{\left(x \right)}\right) \log{\left(4 \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(4 \right)} \tan^{3}{\left(x \right)} + 4 \tan^{2}{\left(x \right)} + 2\right) \log{\left(4 \right)} \tan{\left(x \right)}}{x} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(4 \right)} \tan^{2}{\left(x \right)}}{x^{2}} - \frac{2}{x^{3}}\right) \operatorname{acos}{\left(3 \right)}}{x}
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