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(1+sqrt(x))/(1-sqrt(x))
The derivative of the function/ (1+sqrt(x))/(1-sqrt(x))

Derivative of (1+sqrt(x))/(1-sqrt(x))

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
      ___
1 + \/ x 
---------
      ___
1 - \/ x
x+11−x\frac{\sqrt{x} + 1}{1 - \sqrt{x}} 
  /      ___\
d |1 + \/ x |
--|---------|
dx|      ___|
  \1 - \/ x /
ddxx+11−x\frac{d}{d x} \frac{\sqrt{x} + 1}{1 - \sqrt{x}}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    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)=x+1f{\left(x \right)} = \sqrt{x} + 1 and g(x)=1−xg{\left(x \right)} = 1 - \sqrt{x}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate x+1\sqrt{x} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

      The result is: 12x\frac{1}{2 \sqrt{x}}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate 1−x1 - \sqrt{x} term by term:

      1. The derivative of the constant 11 is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

        So, the result is: −12x- \frac{1}{2 \sqrt{x}}

      The result is: −12x- \frac{1}{2 \sqrt{x}}

    Now plug in to the quotient rule:

    1−x2x+x+12x(1−x)2\frac{\frac{1 - \sqrt{x}}{2 \sqrt{x}} + \frac{\sqrt{x} + 1}{2 \sqrt{x}}}{\left(1 - \sqrt{x}\right)^{2}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
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The first derivative [src] 
                                 ___      
         1                 1 + \/ x       
------------------- + --------------------
    ___ /      ___\                      2
2*\/ x *\1 - \/ x /       ___ /      ___\ 
                      2*\/ x *\1 - \/ x /
12x(1−x)+x+12x(1−x)2\frac{1}{2 \sqrt{x} \left(1 - \sqrt{x}\right)} + \frac{\sqrt{x} + 1}{2 \sqrt{x} \left(1 - \sqrt{x}\right)^{2}}
Simplify
The second derivative [src] 
                        /      ___\ / 1           2       \
                        \1 + \/ x /*|---- + --------------|
                                    | 3/2     /       ___\|
 1           2                      \x      x*\-1 + \/ x //
---- + -------------- - -----------------------------------
 3/2     /       ___\                       ___            
x      x*\-1 + \/ x /                -1 + \/ x             
-----------------------------------------------------------
                         /       ___\                      
                       4*\-1 + \/ x /
−(x+1)(2x(x−1)+1x32)x−1+2x(x−1)+1x324(x−1)\frac{- \frac{\left(\sqrt{x} + 1\right) \left(\frac{2}{x \left(\sqrt{x} - 1\right)} + \frac{1}{x^{\frac{3}{2}}}\right)}{\sqrt{x} - 1} + \frac{2}{x \left(\sqrt{x} - 1\right)} + \frac{1}{x^{\frac{3}{2}}}}{4 \left(\sqrt{x} - 1\right)}
Simplify
The third derivative [src] 
  /                           /      ___\ / 1            2                  2         \                        \
  |                           \1 + \/ x /*|---- + --------------- + ------------------|    1           2       |
  |                                       | 5/2    2 /       ___\                    2|   ---- + --------------|
  |                                       |x      x *\-1 + \/ x /    3/2 /       ___\ |    3/2     /       ___\|
  |   1            1                      \                         x   *\-1 + \/ x / /   x      x*\-1 + \/ x /|
3*|- ---- - --------------- + --------------------------------------------------------- - ---------------------|
  |   5/2    2 /       ___\                                  ___                              ___ /       ___\ |
  \  x      x *\-1 + \/ x /                           -1 + \/ x                             \/ x *\-1 + \/ x / /
----------------------------------------------------------------------------------------------------------------
                                                   /       ___\                                                 
                                                 8*\-1 + \/ x /
3((x+1)(2x2(x−1)+2x32(x−1)2+1x52)x−1−1x2(x−1)−2x(x−1)+1x32x(x−1)−1x52)8(x−1)\frac{3 \left(\frac{\left(\sqrt{x} + 1\right) \left(\frac{2}{x^{2} \left(\sqrt{x} - 1\right)} + \frac{2}{x^{\frac{3}{2}} \left(\sqrt{x} - 1\right)^{2}} + \frac{1}{x^{\frac{5}{2}}}\right)}{\sqrt{x} - 1} - \frac{1}{x^{2} \left(\sqrt{x} - 1\right)} - \frac{\frac{2}{x \left(\sqrt{x} - 1\right)} + \frac{1}{x^{\frac{3}{2}}}}{\sqrt{x} \left(\sqrt{x} - 1\right)} - \frac{1}{x^{\frac{5}{2}}}\right)}{8 \left(\sqrt{x} - 1\right)}
Simplify
The graph
Derivative of (1+sqrt(x))/(1-sqrt(x))
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