Derivada de y=sqrt((x+sqrtx)/(x-sqrtx))

Solución

Ha introducido [src]

      ___________
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    /  x + \/ x  
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\/     x - \/ x

$$\sqrt{\frac{\sqrt{x} + x}{- \sqrt{x} + x}}$$

sqrt((x + sqrt(x))/(x - sqrt(x)))

Solución detallada

Sustituimos $u = \frac{\sqrt{x} + x}{- \sqrt{x} + x}$ .
Según el principio, aplicamos: $\sqrt{u}$ tenemos $\frac{1}{2 \sqrt{u}}$
Luego se aplica una cadena de reglas. Multiplicamos por $\frac{d}{d x} \frac{\sqrt{x} + x}{- \sqrt{x} + x}$ :
1. Se aplica la regla de la derivada parcial:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sqrt{x} + x$ y $g{\left(x \right)} = - \sqrt{x} + x$ .
  
  Para calcular $\frac{d}{d x} f{\left(x \right)}$ :
  1. diferenciamos $\sqrt{x} + x$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. Según el principio, aplicamos: $\sqrt{x}$ tenemos $\frac{1}{2 \sqrt{x}}$
    Como resultado de: $1 + \frac{1}{2 \sqrt{x}}$
  Para calcular $\frac{d}{d x} g{\left(x \right)}$ :
  1. diferenciamos $- \sqrt{x} + x$ miembro por miembro:
    1. Según el principio, aplicamos: $x$ tenemos $1$
    2. La derivada del producto de una constante por función es igual al producto de esta constante por la derivada de esta función.
      1. Según el principio, aplicamos: $\sqrt{x}$ tenemos $\frac{1}{2 \sqrt{x}}$
      Entonces, como resultado: $- \frac{1}{2 \sqrt{x}}$
    Como resultado de: $1 - \frac{1}{2 \sqrt{x}}$
  Ahora aplicamos la regla de la derivada de una divesión:
  
  $\frac{- \left(1 - \frac{1}{2 \sqrt{x}}\right) \left(\sqrt{x} + x\right) + \left(1 + \frac{1}{2 \sqrt{x}}\right) \left(- \sqrt{x} + x\right)}{\left(- \sqrt{x} + x\right)^{2}}$
Como resultado de la secuencia de reglas:

$\frac{- \left(1 - \frac{1}{2 \sqrt{x}}\right) \left(\sqrt{x} + x\right) + \left(1 + \frac{1}{2 \sqrt{x}}\right) \left(- \sqrt{x} + x\right)}{2 \sqrt{\frac{\sqrt{x} + x}{- \sqrt{x} + x}} \left(- \sqrt{x} + x\right)^{2}}$
Simplificamos:

$\frac{x}{2 \sqrt{\frac{- \sqrt{x} - x}{\sqrt{x} - x}} \left(- x^{\frac{5}{2}} - x^{\frac{3}{2}} + 2 x^{2}\right)}$

Respuesta:

$\frac{x}{2 \sqrt{\frac{- \sqrt{x} - x}{\sqrt{x} - x}} \left(- x^{\frac{5}{2}} - x^{\frac{3}{2}} + 2 x^{2}\right)}$

Gráfica

Trazar el gráfico f(x)

Trazar el gráfico f'(x)

Primera derivada [src]

                              /        1       /        1   \ /      ___\\
      ___________             | 1 + -------    |-1 + -------|*\x + \/ x /|
     /       ___              |         ___    |         ___|            |
    /  x + \/ x   /      ___\ |     2*\/ x     \     2*\/ x /            |
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  /          ___              |  /      ___\                      2      |
\/     x - \/ x               |2*\x - \/ x /           /      ___\       |
                              \                      2*\x - \/ x /       /
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                                x + \/ x

$$\frac{\sqrt{\frac{\sqrt{x} + x}{- \sqrt{x} + x}} \left(- \sqrt{x} + x\right) \left(\frac{\left(-1 + \frac{1}{2 \sqrt{x}}\right) \left(\sqrt{x} + x\right)}{2 \left(- \sqrt{x} + x\right)^{2}} + \frac{1 + \frac{1}{2 \sqrt{x}}}{2 \left(- \sqrt{x} + x\right)}\right)}{\sqrt{x} + x}$$

Simplificar

Segunda derivada [src]

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                                                                                                                         16*\x + \/ x /

$$\frac{\sqrt{- \frac{\sqrt{x} + x}{\sqrt{x} - x}} \left(\frac{4 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{4 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{2 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} + x} + \frac{\left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{2}}{\sqrt{x} + x} - \frac{2}{x^{\frac{3}{2}}} + \frac{2 \left(\sqrt{x} + x\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{16 \left(\sqrt{x} + x\right)}$$

Simplificar

Tercera derivada [src]

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                                                                                                                                                                                                                                                                                                                                                                                                                                       64*\x + \/ x /

$$\frac{\sqrt{- \frac{\sqrt{x} + x}{\sqrt{x} - x}} \left(\frac{24 \left(2 - \frac{1}{\sqrt{x}}\right)^{3} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{3}} + \frac{24 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(2 + \frac{1}{\sqrt{x}}\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{8 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\left(\sqrt{x} - x\right) \left(\sqrt{x} + x\right)} - \frac{8 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\frac{2 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{1}{x^{\frac{3}{2}}} + \frac{\sqrt{x} + x}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{\sqrt{x} - x} - \frac{6 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{2}}{\left(\sqrt{x} - x\right) \left(\sqrt{x} + x\right)} + \frac{8 \left(2 + \frac{1}{\sqrt{x}}\right)^{2} \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{\left(\sqrt{x} + x\right)^{2}} - \frac{8 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{2 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{1}{x^{\frac{3}{2}}} + \frac{\sqrt{x} + x}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{\sqrt{x} + x} - \frac{6 \left(2 + \frac{1}{\sqrt{x}}\right) \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{2}}{\left(\sqrt{x} + x\right)^{2}} + \frac{6 \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right) \left(\frac{2 \left(2 - \frac{1}{\sqrt{x}}\right)^{2} \left(\sqrt{x} + x\right)}{\left(\sqrt{x} - x\right)^{2}} + \frac{2 \left(2 - \frac{1}{\sqrt{x}}\right) \left(2 + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} - x} - \frac{1}{x^{\frac{3}{2}}} + \frac{\sqrt{x} + x}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)}\right)}{\sqrt{x} + x} + \frac{\left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)^{3}}{\left(\sqrt{x} + x\right)^{2}} - \frac{12 \left(2 - \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)} + \frac{24 \left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)^{2}} + \frac{12 \left(2 + \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)} + \frac{4 \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} + x\right)} - \frac{4 \left(\frac{\left(2 - \frac{1}{\sqrt{x}}\right) \left(\sqrt{x} + x\right)}{\sqrt{x} - x} + 2 + \frac{1}{\sqrt{x}}\right)}{x^{\frac{3}{2}} \left(\sqrt{x} - x\right)} + \frac{12}{x^{\frac{5}{2}}} - \frac{12 \left(\sqrt{x} + x\right)}{x^{\frac{5}{2}} \left(\sqrt{x} - x\right)}\right)}{64 \left(\sqrt{x} + x\right)}$$

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Derivada de y=sqrt((x+sqrtx)/(x-sqrtx))

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Solución