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xtanxsqrt
Derivada de la función/ xsqrt/ xtanx/ xtanxsqrt

Derivada de xtanxsqrt

Función f() - derivada -er orden en el punto
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Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
           ___
x*tan(x)*\/ x
xxtan⁡(x)\sqrt{x} x \tan{\left(x \right)} 
(x*tan(x))*sqrt(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)=xtan⁡(x)f{\left(x \right)} = x \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)=xf{\left(x \right)} = x; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: xx tenemos 11

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

    g(x)=xg{\left(x \right)} = \sqrt{x}; calculamos ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Según el principio, aplicamos: x\sqrt{x} tenemos 12x\frac{1}{2 \sqrt{x}}

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

  2. Simplificamos:

    x(x+3sin⁡(2x)4)cos⁡2(x)\frac{\sqrt{x} \left(x + \frac{3 \sin{\left(2 x \right)}}{4}\right)}{\cos^{2}{\left(x \right)}}

Respuesta:

x(x+3sin⁡(2x)4)cos⁡2(x)\frac{\sqrt{x} \left(x + \frac{3 \sin{\left(2 x \right)}}{4}\right)}{\cos^{2}{\left(x \right)}}

Gráfica
02468-8-6-4-2-1010-1000010000
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Primera derivada [src] 
                                     ___       
  ___ /  /       2   \         \   \/ x *tan(x)
\/ x *\x*\1 + tan (x)/ + tan(x)/ + ------------
                                        2
x(x(tan⁡2(x)+1)+tan⁡(x))+xtan⁡(x)2\sqrt{x} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right) + \frac{\sqrt{x} \tan{\left(x \right)}}{2}
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Segunda derivada [src] 
  /       2   \                                                                    
x*\1 + tan (x)/ + tan(x)       ___ /       2        /       2   \       \    tan(x)
------------------------ + 2*\/ x *\1 + tan (x) + x*\1 + tan (x)/*tan(x)/ - -------
           ___                                                                  ___
         \/ x                                                               4*\/ x
2x(x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1)+x(tan⁡2(x)+1)+tan⁡(x)x−tan⁡(x)4x2 \sqrt{x} \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) + \frac{x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}}{\sqrt{x}} - \frac{\tan{\left(x \right)}}{4 \sqrt{x}}
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Tercera derivada [src] 
  /       2        /       2   \       \     /  /       2   \         \                                                                  
3*\1 + tan (x) + x*\1 + tan (x)/*tan(x)/   3*\x*\1 + tan (x)/ + tan(x)/   3*tan(x)       ___ /       2   \ /             /         2   \\
---------------------------------------- - ---------------------------- + -------- + 2*\/ x *\1 + tan (x)/*\3*tan(x) + x*\1 + 3*tan (x)//
                   ___                                   3/2                  3/2                                                        
                 \/ x                                 4*x                  8*x
2x(x(3tan⁡2(x)+1)+3tan⁡(x))(tan⁡2(x)+1)+3(x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1)x−3(x(tan⁡2(x)+1)+tan⁡(x))4x32+3tan⁡(x)8x322 \sqrt{x} \left(x \left(3 \tan^{2}{\left(x \right)} + 1\right) + 3 \tan{\left(x \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right)}{\sqrt{x}} - \frac{3 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)}\right)}{4 x^{\frac{3}{2}}} + \frac{3 \tan{\left(x \right)}}{8 x^{\frac{3}{2}}}
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Gráfico
Derivada de xtanxsqrt
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