Derivative of ln(√x+√(x+1))

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   /  ___     _______\
log\\/ x  + \/ x + 1 /

$$\log{\left(\sqrt{x} + \sqrt{x + 1} \right)}$$

log(sqrt(x) + sqrt(x + 1))

Detail solution

Let $u = \sqrt{x} + \sqrt{x + 1}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\sqrt{x} + \sqrt{x + 1}\right)$ :
1. Differentiate $\sqrt{x} + \sqrt{x + 1}$ term by term:
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  2. Let $u = x + 1$ .
  3. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
    1. Differentiate $x + 1$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of the constant $1$ is zero.
      The result is: $1$
    The result of the chain rule is:
    
    $\frac{1}{2 \sqrt{x + 1}}$
  The result is: $\frac{1}{2 \sqrt{x + 1}} + \frac{1}{2 \sqrt{x}}$
The result of the chain rule is:

$\frac{\frac{1}{2 \sqrt{x + 1}} + \frac{1}{2 \sqrt{x}}}{\sqrt{x} + \sqrt{x + 1}}$
Now simplify:

$\frac{1}{2 \sqrt{x} \sqrt{x + 1}}$

The answer is:

$\frac{1}{2 \sqrt{x} \sqrt{x + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1           1     
------- + -----------
    ___       _______
2*\/ x    2*\/ x + 1 
---------------------
    ___     _______  
  \/ x  + \/ x + 1

$$\frac{\frac{1}{2 \sqrt{x + 1}} + \frac{1}{2 \sqrt{x}}}{\sqrt{x} + \sqrt{x + 1}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                         3                                            
                      /  1         1    \      / 1         1     \ /  1         1    \
                    2*|----- + ---------|    3*|---- + ----------|*|----- + ---------|
                      |  ___     _______|      | 3/2          3/2| |  ___     _______|
 3         3          \\/ x    \/ 1 + x /      \x      (1 + x)   / \\/ x    \/ 1 + x /
---- + ---------- + ---------------------- + -----------------------------------------
 5/2          5/2                       2                  ___     _______            
x      (1 + x)       /  ___     _______\                 \/ x  + \/ 1 + x             
                     \\/ x  + \/ 1 + x /                                              
--------------------------------------------------------------------------------------
                                  /  ___     _______\                                 
                                8*\\/ x  + \/ 1 + x /

$$\frac{\frac{3}{\left(x + 1\right)^{\frac{5}{2}}} + \frac{3 \left(\frac{1}{\left(x + 1\right)^{\frac{3}{2}}} + \frac{1}{x^{\frac{3}{2}}}\right) \left(\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}\right)}{\sqrt{x} + \sqrt{x + 1}} + \frac{2 \left(\frac{1}{\sqrt{x + 1}} + \frac{1}{\sqrt{x}}\right)^{3}}{\left(\sqrt{x} + \sqrt{x + 1}\right)^{2}} + \frac{3}{x^{\frac{5}{2}}}}{8 \left(\sqrt{x} + \sqrt{x + 1}\right)}$$

Simplify

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Derivative of ln(√x+√(x+1))

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The solution