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y=x^2tg(x)*sin(x)+1
Derivada de la función/ y=x^2/ tg(x)/ sin(x)/ y=x^2tg(x)*sin(x)+1

Derivada de y=x^2tg(x)*sin(x)+1

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

Ha introducido [src] 
 2                  
x *tan(x)*sin(x) + 1
x2tan⁡(x)sin⁡(x)+1x^{2} \tan{\left(x \right)} \sin{\left(x \right)} + 1 
(x^2*tan(x))*sin(x) + 1
Solución detallada

  1. diferenciamos x2tan⁡(x)sin⁡(x)+1x^{2} \tan{\left(x \right)} \sin{\left(x \right)} + 1 miembro por miembro:

    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)=x2tan⁡(x)f{\left(x \right)} = x^{2} \tan{\left(x \right)}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      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)=x2f{\left(x \right)} = x^{2}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Según el principio, aplicamos: x2x^{2} tenemos 2x2 x

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

        Como resultado de: x2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+2xtan⁡(x)\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x \tan{\left(x \right)}

      g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}; calculamos ddxg(x)\frac{d}{d x} g{\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)}

      Como resultado de: x2cos⁡(x)tan⁡(x)+(x2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+2xtan⁡(x))sin⁡(x)x^{2} \cos{\left(x \right)} \tan{\left(x \right)} + \left(\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x \tan{\left(x \right)}\right) \sin{\left(x \right)}

    2. La derivada de una constante 11 es igual a cero.

    Como resultado de: x2cos⁡(x)tan⁡(x)+(x2(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+2xtan⁡(x))sin⁡(x)x^{2} \cos{\left(x \right)} \tan{\left(x \right)} + \left(\frac{x^{2} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + 2 x \tan{\left(x \right)}\right) \sin{\left(x \right)}

  2. Simplificamos:

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

Respuesta:

x(x+xcos⁡2(x)+2tan⁡(x))sin⁡(x)x \left(x + \frac{x}{\cos^{2}{\left(x \right)}} + 2 \tan{\left(x \right)}\right) \sin{\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 /       2   \             \           2              
\x *\1 + tan (x)/ + 2*x*tan(x)/*sin(x) + x *cos(x)*tan(x)
x2cos⁡(x)tan⁡(x)+(x2(tan⁡2(x)+1)+2xtan⁡(x))sin⁡(x)x^{2} \cos{\left(x \right)} \tan{\left(x \right)} + \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) + 2 x \tan{\left(x \right)}\right) \sin{\left(x \right)}
Simplificar
Segunda derivada [src] 
  /    /       2   \    2 /       2   \                \            /             /       2   \\           2 /       2   \           2                                  
2*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/*sin(x) + x*\2*tan(x) + x*\1 + tan (x)//*cos(x) + x *\1 + tan (x)/*cos(x) - x *sin(x)*tan(x) + 2*x*cos(x)*tan(x)
x2(tan⁡2(x)+1)cos⁡(x)−x2sin⁡(x)tan⁡(x)+x(x(tan⁡2(x)+1)+2tan⁡(x))cos⁡(x)+2xcos⁡(x)tan⁡(x)+2(x2(tan⁡2(x)+1)tan⁡(x)+2x(tan⁡2(x)+1)+tan⁡(x))sin⁡(x)x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - x^{2} \sin{\left(x \right)} \tan{\left(x \right)} + x \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \cos{\left(x \right)} + 2 x \cos{\left(x \right)} \tan{\left(x \right)} + 2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \sin{\left(x \right)}
Simplificar
Tercera derivada [src] 
  /                                2                                                        \                                                                                                                                                                                                                                                                    
  |         2       2 /       2   \       2    2    /       2   \       /       2   \       |                              /    /       2   \    2 /       2   \                \            /             /       2   \\           2                                        2 /       2   \              /       2   \             2 /       2   \              
2*\3 + 3*tan (x) + x *\1 + tan (x)/  + 2*x *tan (x)*\1 + tan (x)/ + 6*x*\1 + tan (x)/*tan(x)/*sin(x) + 2*cos(x)*tan(x) + 4*\2*x*\1 + tan (x)/ + x *\1 + tan (x)/*tan(x) + tan(x)/*cos(x) - x*\2*tan(x) + x*\1 + tan (x)//*sin(x) - x *cos(x)*tan(x) - 4*x*sin(x)*tan(x) - 2*x *\1 + tan (x)/*sin(x) + 4*x*\1 + tan (x)/*cos(x) + 2*x *\1 + tan (x)/*cos(x)*tan(x)
−2x2(tan⁡2(x)+1)sin⁡(x)+2x2(tan⁡2(x)+1)cos⁡(x)tan⁡(x)−x2cos⁡(x)tan⁡(x)−x(x(tan⁡2(x)+1)+2tan⁡(x))sin⁡(x)+4x(tan⁡2(x)+1)cos⁡(x)−4xsin⁡(x)tan⁡(x)+4(x2(tan⁡2(x)+1)tan⁡(x)+2x(tan⁡2(x)+1)+tan⁡(x))cos⁡(x)+2(x2(tan⁡2(x)+1)2+2x2(tan⁡2(x)+1)tan⁡2(x)+6x(tan⁡2(x)+1)tan⁡(x)+3tan⁡2(x)+3)sin⁡(x)+2cos⁡(x)tan⁡(x)- 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} \tan{\left(x \right)} - x^{2} \cos{\left(x \right)} \tan{\left(x \right)} - x \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)}\right) \sin{\left(x \right)} + 4 x \left(\tan^{2}{\left(x \right)} + 1\right) \cos{\left(x \right)} - 4 x \sin{\left(x \right)} \tan{\left(x \right)} + 4 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 2 x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) \cos{\left(x \right)} + 2 \left(x^{2} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 2 x^{2} \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 6 x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + 3 \tan^{2}{\left(x \right)} + 3\right) \sin{\left(x \right)} + 2 \cos{\left(x \right)} \tan{\left(x \right)}
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Gráfico
Derivada de y=x^2tg(x)*sin(x)+1
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