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y=tgx^3*cosx
Derivada de la función/ y=tgx/ x^3*cosx/ y=tgx^3*cosx

Derivada de y=tgx^3*cosx

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

Ha introducido [src] 
   3          
tan (x)*cos(x)
cos⁡(x)tan⁡3(x)\cos{\left(x \right)} \tan^{3}{\left(x \right)} 
tan(x)^3*cos(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)=tan⁡3(x)f{\left(x \right)} = \tan^{3}{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\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)}}

    g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}; calculamos 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)}

    Como resultado de: 3(sin⁡2(x)+cos⁡2(x))tan⁡2(x)cos⁡(x)−sin⁡(x)tan⁡3(x)\frac{3 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{2}{\left(x \right)}}{\cos{\left(x \right)}} - \sin{\left(x \right)} \tan^{3}{\left(x \right)}

  2. Simplificamos:

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

Respuesta:

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

Gráfica
02468-8-6-4-2-1010-5000050000
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Trazar el gráfico f'(x)
Primera derivada [src] 
     3                2    /         2   \       
- tan (x)*sin(x) + tan (x)*\3 + 3*tan (x)/*cos(x)
(3tan⁡2(x)+3)cos⁡(x)tan⁡2(x)−sin⁡(x)tan⁡3(x)\left(3 \tan^{2}{\left(x \right)} + 3\right) \cos{\left(x \right)} \tan^{2}{\left(x \right)} - \sin{\left(x \right)} \tan^{3}{\left(x \right)}
Simplificar
Segunda derivada [src] 
/     2               /       2   \                   /       2   \ /         2   \       \       
\- tan (x)*cos(x) - 6*\1 + tan (x)/*sin(x)*tan(x) + 6*\1 + tan (x)/*\1 + 2*tan (x)/*cos(x)/*tan(x)
(6(tan⁡2(x)+1)(2tan⁡2(x)+1)cos⁡(x)−6(tan⁡2(x)+1)sin⁡(x)tan⁡(x)−cos⁡(x)tan⁡2(x))tan⁡(x)\left(6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - 6 \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} - \cos{\left(x \right)} \tan^{2}{\left(x \right)}\right) \tan{\left(x \right)}
Simplificar
Tercera derivada [src] 
                                                                  /             2                                      \                                                        
   3                  2    /       2   \            /       2   \ |/       2   \         4           2    /       2   \|             /       2   \ /         2   \              
tan (x)*sin(x) - 9*tan (x)*\1 + tan (x)/*cos(x) + 6*\1 + tan (x)/*\\1 + tan (x)/  + 2*tan (x) + 7*tan (x)*\1 + tan (x)//*cos(x) - 18*\1 + tan (x)/*\1 + 2*tan (x)/*sin(x)*tan(x)
−18(tan⁡2(x)+1)(2tan⁡2(x)+1)sin⁡(x)tan⁡(x)+6(tan⁡2(x)+1)((tan⁡2(x)+1)2+7(tan⁡2(x)+1)tan⁡2(x)+2tan⁡4(x))cos⁡(x)−9(tan⁡2(x)+1)cos⁡(x)tan⁡2(x)+sin⁡(x)tan⁡3(x)- 18 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} \tan{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 7 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 2 \tan^{4}{\left(x \right)}\right) \cos{\left(x \right)} - 9 \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan^{2}{\left(x \right)} + \sin{\left(x \right)} \tan^{3}{\left(x \right)}
Simplificar
Gráfico
Derivada de y=tgx^3*cosx
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