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(x+x^1/2)/tan(x)*x^5

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  • Derivada de:
  • Derivada de x^-4/5 Derivada de x^-4/5
  • Derivada de x^2*5^x Derivada de x^2*5^x
  • Derivada de x/(1+e^x) Derivada de x/(1+e^x)
  • Derivada de u/v

  • Expresiones idénticas

  • (x+x^ uno / dos)/tan(x)*x^ cinco
  • (x más x en el grado 1 dividir por 2) dividir por tangente de (x) multiplicar por x en el grado 5
  • (x más x en el grado uno dividir por dos) dividir por tangente de (x) multiplicar por x en el grado cinco
  • (x+x1/2)/tan(x)*x5
  • x+x1/2/tanx*x5
  • (x+x^1/2)/tan(x)*x⁵
  • (x+x^1/2)/tan(x)x^5
  • (x+x1/2)/tan(x)x5
  • x+x1/2/tanxx5
  • x+x^1/2/tanxx^5
  • (x+x^1 dividir por 2) dividir por tan(x)*x^5

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  • (x-x^1/2)/tan(x)*x^5
Derivada de la función/ x^1/2/ tan(x)/ (x+x^1/2)/tan(x)*x^5

Derivada de (x+x^1/2)/tan(x)*x^5

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

Gráfico:

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Definida a trozos:

Solución

Ha introducido [src] 
      ___   
x + \/ x   5
---------*x 
  tan(x)
x5x+xtan⁡(x)x^{5} \frac{\sqrt{x} + x}{\tan{\left(x \right)}} 
((x + sqrt(x))/tan(x))*x^5
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)=x5(x+x)f{\left(x \right)} = x^{5} \left(\sqrt{x} + x\right) 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. 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)=x5f{\left(x \right)} = x^{5}; calculamos ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Según el principio, aplicamos: x5x^{5} tenemos 5x45 x^{4}

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

      1. diferenciamos x+x\sqrt{x} + x miembro por miembro:

        1. Según el principio, aplicamos: xx tenemos 11

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

        Como resultado de: 1+12x1 + \frac{1}{2 \sqrt{x}}

      Como resultado de: x5(1+12x)+5x4(x+x)x^{5} \left(1 + \frac{1}{2 \sqrt{x}}\right) + 5 x^{4} \left(\sqrt{x} + x\right)

    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:

    −x5(x+x)(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+(x5(1+12x)+5x4(x+x))tan⁡(x)tan⁡2(x)\frac{- \frac{x^{5} \left(\sqrt{x} + x\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \left(x^{5} \left(1 + \frac{1}{2 \sqrt{x}}\right) + 5 x^{4} \left(\sqrt{x} + x\right)\right) \tan{\left(x \right)}}{\tan^{2}{\left(x \right)}}

  2. Simplificamos:

    −4x112+11x92sin⁡(2x)−4x6+12x5sin⁡(2x)2(1−cos⁡(2x))\frac{- 4 x^{\frac{11}{2}} + 11 x^{\frac{9}{2}} \sin{\left(2 x \right)} - 4 x^{6} + 12 x^{5} \sin{\left(2 x \right)}}{2 \left(1 - \cos{\left(2 x \right)}\right)}

Respuesta:

−4x112+11x92sin⁡(2x)−4x6+12x5sin⁡(2x)2(1−cos⁡(2x))\frac{- 4 x^{\frac{11}{2}} + 11 x^{\frac{9}{2}} \sin{\left(2 x \right)} - 4 x^{6} + 12 x^{5} \sin{\left(2 x \right)}}{2 \left(1 - \cos{\left(2 x \right)}\right)}

Gráfica
02468-8-6-4-2-1010-10000000001000000000
Trazar el gráfico f(x)
Trazar el gráfico f'(x)
Primera derivada [src] 
   /       1                                \                   
   |1 + -------                             |                   
   |        ___   /        2   \ /      ___\|      4 /      ___\
 5 |    2*\/ x    \-1 - tan (x)/*\x + \/ x /|   5*x *\x + \/ x /
x *|----------- + --------------------------| + ----------------
   |   tan(x)                 2             |        tan(x)     
   \                       tan (x)          /
x5(1+12xtan⁡(x)+(x+x)(−tan⁡2(x)−1)tan⁡2(x))+5x4(x+x)tan⁡(x)x^{5} \left(\frac{1 + \frac{1}{2 \sqrt{x}}}{\tan{\left(x \right)}} + \frac{\left(\sqrt{x} + x\right) \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}}\right) + \frac{5 x^{4} \left(\sqrt{x} + x\right)}{\tan{\left(x \right)}}
Simplificar
Segunda derivada [src] 
   /                                                                     /                                                          /       2   \ /      1  \\\
   |                                                                     |                                                        4*\1 + tan (x)/*|2 + -----|||
   |                                                                     |                       /            2   \                               |      ___|||
   |                                                                   2 | 1       /       2   \ |     1 + tan (x)| /      ___\                   \    \/ x /||
   |                                                                  x *|---- - 8*\1 + tan (x)/*|-1 + -----------|*\x + \/ x / + ---------------------------||
   |                      /              /       2   \ /      ___\\      | 3/2                   |          2     |                          tan(x)          ||
 3 |            ___       |      1     2*\1 + tan (x)/*\x + \/ x /|      \x                      \       tan (x)  /                                          /|
x *|20*x + 20*\/ x  + 5*x*|2 + ----- - ---------------------------| - ----------------------------------------------------------------------------------------|
   |                      |      ___              tan(x)          |                                              4                                            |
   \                      \    \/ x                               /                                                                                           /
---------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                             tan(x)
x3(20x−x2(4(2+1x)(tan⁡2(x)+1)tan⁡(x)−8(x+x)(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)+1x32)4+5x(−2(x+x)(tan⁡2(x)+1)tan⁡(x)+2+1x)+20x)tan⁡(x)\frac{x^{3} \left(20 \sqrt{x} - \frac{x^{2} \left(\frac{4 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 8 \left(\sqrt{x} + x\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{1}{x^{\frac{3}{2}}}\right)}{4} + 5 x \left(- \frac{2 \left(\sqrt{x} + x\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 2 + \frac{1}{\sqrt{x}}\right) + 20 x\right)}{\tan{\left(x \right)}}
Simplificar
Tercera derivada [src] 
   /                    /                                                                                                                          /            2   \            \                                                                                                                                               \
   |                    |                                                                                                            /       2   \ |     1 + tan (x)| /      1  \|                                                          /                                                          /       2   \ /      1  \\|
   |                    |                 /                               2                  3\                                   24*\1 + tan (x)/*|-1 + -----------|*|2 + -----||                                                          |                                                        4*\1 + tan (x)/*|2 + -----|||
   |                    |                 |                  /       2   \      /       2   \ |                   /       2   \                    |          2     | |      ___||        /              /       2   \ /      ___\\         |                       /            2   \                               |      ___|||
   |                  3 |     /      ___\ |         2      5*\1 + tan (x)/    3*\1 + tan (x)/ |        3        6*\1 + tan (x)/                    \       tan (x)  / \    \/ x /|        |      1     2*\1 + tan (x)/*\x + \/ x /|       2 | 1       /       2   \ |     1 + tan (x)| /      ___\                   \    \/ x /||
   |                 x *|- 16*\x + \/ x /*|2 + 2*tan (x) - ---------------- + ----------------| + ----------- + --------------- + -----------------------------------------------|   30*x*|2 + ----- - ---------------------------|   15*x *|---- - 8*\1 + tan (x)/*|-1 + -----------|*\x + \/ x / + ---------------------------||
   |   /      ___\      |                 |                       2                  4        |    5/2             3/2    2                            tan(x)                    |        |      ___              tan(x)          |         | 3/2                   |          2     |                          tan(x)          ||
 2 |60*\x + \/ x /      \                 \                    tan (x)            tan (x)     /   x   *tan(x)     x   *tan (x)                                                   /        \    \/ x                               /         \x                      \       tan (x)  /                                          /|
x *|-------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------- - -------------------------------------------------------------------------------------------|
   \    tan(x)                                                                                     8                                                                                                     tan(x)                                                                 4*tan(x)                                         /
x2(x3(24(2+1x)(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)tan⁡(x)−16(x+x)(3(tan⁡2(x)+1)3tan⁡4(x)−5(tan⁡2(x)+1)2tan⁡2(x)+2tan⁡2(x)+2)+6(tan⁡2(x)+1)x32tan⁡2(x)+3x52tan⁡(x))8−15x2(4(2+1x)(tan⁡2(x)+1)tan⁡(x)−8(x+x)(tan⁡2(x)+1tan⁡2(x)−1)(tan⁡2(x)+1)+1x32)4tan⁡(x)+30x(−2(x+x)(tan⁡2(x)+1)tan⁡(x)+2+1x)tan⁡(x)+60(x+x)tan⁡(x))x^{2} \left(\frac{x^{3} \left(\frac{24 \left(2 + \frac{1}{\sqrt{x}}\right) \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)}} - 16 \left(\sqrt{x} + x\right) \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{6 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{\frac{3}{2}} \tan^{2}{\left(x \right)}} + \frac{3}{x^{\frac{5}{2}} \tan{\left(x \right)}}\right)}{8} - \frac{15 x^{2} \left(\frac{4 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} - 8 \left(\sqrt{x} + x\right) \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{1}{x^{\frac{3}{2}}}\right)}{4 \tan{\left(x \right)}} + \frac{30 x \left(- \frac{2 \left(\sqrt{x} + x\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}} + 2 + \frac{1}{\sqrt{x}}\right)}{\tan{\left(x \right)}} + \frac{60 \left(\sqrt{x} + x\right)}{\tan{\left(x \right)}}\right)
Simplificar
Gráfico
Derivada de (x+x^1/2)/tan(x)*x^5
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