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Graphing y =:
log(x+sqrt(x^2+1))
Identical expressions
log(x+sqrt(x^ two + one))
logarithm of (x plus square root of (x squared plus 1))
logarithm of (x plus square root of (x to the power of two plus one))
log(x+√(x^2+1))
log(x+sqrt(x2+1))
logx+sqrtx2+1
log(x+sqrt(x²+1))
log(x+sqrt(x to the power of 2+1))
logx+sqrtx^2+1
Similar expressions
log(x+sqrt(x^2-1))
log(x-sqrt(x^2+1))

The derivative of the function/ log(x+sqrt(x^2+1))

Derivative of log(x+sqrt(x^2+1))

The solution

You have entered [src]

   /       ________\
   |      /  2     |
log\x + \/  x  + 1 /

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

  /   /       ________\\
d |   |      /  2     ||
--\log\x + \/  x  + 1 //
dx

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         x     
1 + -----------
       ________
      /  2     
    \/  x  + 1 
---------------
       ________
      /  2     
x + \/  x  + 1

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

Simplify

The second derivative [src]

 /                               2\ 
 |        2     /         x     \ | 
 |       x      |1 + -----------| | 
 |-1 + ------   |       ________| | 
 |          2   |      /      2 | | 
 |     1 + x    \    \/  1 + x  / | 
-|----------- + ------------------| 
 |   ________           ________  | 
 |  /      2           /      2   | 
 \\/  1 + x      x + \/  1 + x    / 
------------------------------------
                 ________           
                /      2            
          x + \/  1 + x

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

Simplify

The third derivative [src]

                   3                                           /        2  \
  /         x     \        /        2  \     /         x     \ |       x   |
2*|1 + -----------|        |       x   |   3*|1 + -----------|*|-1 + ------|
  |       ________|    3*x*|-1 + ------|     |       ________| |          2|
  |      /      2 |        |          2|     |      /      2 | \     1 + x /
  \    \/  1 + x  /        \     1 + x /     \    \/  1 + x  /              
-------------------- + ----------------- + ---------------------------------
                  2               3/2           ________ /       ________\  
 /       ________\        /     2\             /      2  |      /      2 |  
 |      /      2 |        \1 + x /           \/  1 + x  *\x + \/  1 + x  /  
 \x + \/  1 + x  /                                                          
----------------------------------------------------------------------------
                                     ________                               
                                    /      2                                
                              x + \/  1 + x

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

Simplify

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Derivative of log(x+sqrt(x^2+1))

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The solution