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How do you in partial fractions?:
-1/(2*sqrt(x)*(1+x))
Identical expressions
- one /(two *sqrt(x)*(one +x))
minus 1 divide by (2 multiply by square root of (x) multiply by (1 plus x))
minus one divide by (two multiply by square root of (x) multiply by (one plus x))
-1/(2*√(x)*(1+x))
-1/(2sqrt(x)(1+x))
-1/2sqrtx1+x
-1 divide by (2*sqrt(x)*(1+x))
Similar expressions
1/(2*sqrt(x)*(1+x))
-1/(2*sqrt(x)*(1-x))

The derivative of the function/ -1/(2*sqrt(x)*(1+x))

Derivative of -1/(2sqrt(x)(1+x))

The solution

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      -1       
---------------
    ___        
2*\/ x *(1 + x)

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

d /      -1       \
--|---------------|
dx|    ___        |
  \2*\/ x *(1 + x)/

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

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Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = 2 \sqrt{x} \left(x + 1\right)$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 \sqrt{x} \left(x + 1\right)$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      $g{\left(x \right)} = x + 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Differentiate $x + 1$ term by term:
        
        The derivative of the constant $1$ is zero.
        
        Apply the power rule: $x$ goes to $1$
        
        The result is: $1$
      The result is: $\sqrt{x} + \frac{x + 1}{2 \sqrt{x}}$
    So, the result is: $2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}$
  The result of the chain rule is:
  
  $- \frac{2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}}{4 x \left(x + 1\right)^{2}}$
So, the result is: $\frac{2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}}{4 x \left(x + 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

 /      ___   1 + x\ 
-|- 2*\/ x  - -----| 
 |              ___| 
 \            \/ x / 
---------------------
                2    
     4*x*(1 + x)

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

Simplify

The second derivative [src]

 /    ___   1 + x               /1     2  \ /    ___   1 + x\     /    ___   1 + x\\ 
 |2*\/ x  + -----       1 + x   |- + -----|*|2*\/ x  + -----|   2*|2*\/ x  + -----|| 
 |            ___   4 - -----   \x   1 + x/ |            ___|     |            ___|| 
 |          \/ x          x                 \          \/ x /     \          \/ x /| 
-|--------------- - --------- + ----------------------------- + -------------------| 
 |        2             3/2                   x                      x*(1 + x)     | 
 \       x             x                                                           / 
-------------------------------------------------------------------------------------
                                               2                                     
                                      8*(1 + x)

$$- \frac{\frac{\left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right) \left(\frac{2}{x + 1} + \frac{1}{x}\right)}{x} + \frac{2 \cdot \left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right)}{x \left(x + 1\right)} + \frac{2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}}{x^{2}} - \frac{4 - \frac{x + 1}{x}}{x^{\frac{3}{2}}}}{8 \left(x + 1\right)^{2}}$$

Simplify

The third derivative [src]

                                    /    ___   1 + x\   /    ___   1 + x\ /3       8           4    \   /1     2  \ /    ___   1 + x\                                               /    ___   1 + x\      /    ___   1 + x\     /1     2  \ /    ___   1 + x\
    /    1 + x\     /    1 + x\   4*|2*\/ x  + -----|   |2*\/ x  + -----|*|-- + -------- + ---------|   |- + -----|*|2*\/ x  + -----|   /    1 + x\ /1     2  \     /    1 + x\   8*|2*\/ x  + -----|   12*|2*\/ x  + -----|   2*|- + -----|*|2*\/ x  + -----|
  3*|2 - -----|   3*|4 - -----|     |            ___|   |            ___| | 2          2   x*(1 + x)|   \x   1 + x/ |            ___|   |4 - -----|*|- + -----|   6*|4 - -----|     |            ___|      |            ___|     \x   1 + x/ |            ___|
    \      x  /     \      x  /     \          \/ x /   \          \/ x / \x    (1 + x)             /               \          \/ x /   \      x  / \x   1 + x/     \      x  /     \          \/ x /      \          \/ x /                 \          \/ x /
- ------------- - ------------- + ------------------- + --------------------------------------------- + ----------------------------- - ----------------------- - ------------- + ------------------- + -------------------- + -------------------------------
        5/2             5/2                 3                                 x                                        2                           3/2              3/2                 2                             2                   x*(1 + x)           
       x               x                   x                                                                          x                           x                x   *(1 + x)        x *(1 + x)            x*(1 + x)                                        
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                   2                                                                                                                          
                                                                                                                         16*(1 + x)

$$\frac{\frac{\left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right) \left(\frac{8}{\left(x + 1\right)^{2}} + \frac{4}{x \left(x + 1\right)} + \frac{3}{x^{2}}\right)}{x} + \frac{2 \cdot \left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right) \left(\frac{2}{x + 1} + \frac{1}{x}\right)}{x \left(x + 1\right)} + \frac{\left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right) \left(\frac{2}{x + 1} + \frac{1}{x}\right)}{x^{2}} - \frac{\left(4 - \frac{x + 1}{x}\right) \left(\frac{2}{x + 1} + \frac{1}{x}\right)}{x^{\frac{3}{2}}} + \frac{12 \cdot \left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right)}{x \left(x + 1\right)^{2}} + \frac{8 \cdot \left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right)}{x^{2} \left(x + 1\right)} + \frac{4 \cdot \left(2 \sqrt{x} + \frac{x + 1}{\sqrt{x}}\right)}{x^{3}} - \frac{6 \cdot \left(4 - \frac{x + 1}{x}\right)}{x^{\frac{3}{2}} \left(x + 1\right)} - \frac{3 \cdot \left(2 - \frac{x + 1}{x}\right)}{x^{\frac{5}{2}}} - \frac{3 \cdot \left(4 - \frac{x + 1}{x}\right)}{x^{\frac{5}{2}}}}{16 \left(x + 1\right)^{2}}$$

Simplify

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Derivative of -1/(2sqrt(x)(1+x))

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The solution

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Derivative of -1/(2*sqrt(x)*(1+x))

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Piecewise:

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Derivative of -1/(2sqrt(x)(1+x))