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

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /       ________\
   |      /      2 |
log\x + \/  1 + x  /
log⁡(x+x2+1)\log{\left(x + \sqrt{x^{2} + 1} \right)} 
  /   /       ________\\
d |   |      /      2 ||
--\log\x + \/  1 + x  //
dx
ddxlog⁡(x+x2+1)\frac{d}{d x} \log{\left(x + \sqrt{x^{2} + 1} \right)}
Transform
Detail solution

  1. Let u=x+x2+1u = x + \sqrt{x^{2} + 1}.

  2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

  3. Then, apply the chain rule. Multiply by ddx(x+x2+1)\frac{d}{d x} \left(x + \sqrt{x^{2} + 1}\right):

    1. Differentiate x+x2+1x + \sqrt{x^{2} + 1} term by term:

      1. Apply the power rule: xx goes to 11

      2. Let u=x2+1u = x^{2} + 1.

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

      4. Then, apply the chain rule. Multiply by ddx(x2+1)\frac{d}{d x} \left(x^{2} + 1\right):

        1. Differentiate x2+1x^{2} + 1 term by term:

          1. The derivative of the constant 11 is zero.

          2. Apply the power rule: x2x^{2} goes to 2x2 x

          The result is: 2x2 x

        The result of the chain rule is:

        xx2+1\frac{x}{\sqrt{x^{2} + 1}}

      The result is: xx2+1+1\frac{x}{\sqrt{x^{2} + 1}} + 1

    The result of the chain rule is:

    xx2+1+1x+x2+1\frac{\frac{x}{\sqrt{x^{2} + 1}} + 1}{x + \sqrt{x^{2} + 1}}

  4. Now simplify:

    1x2+1\frac{1}{\sqrt{x^{2} + 1}}

The answer is:

1x2+1\frac{1}{\sqrt{x^{2} + 1}}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
         x     
1 + -----------
       ________
      /      2 
    \/  1 + x  
---------------
       ________
      /      2 
x + \/  1 + x
xx2+1+1x+x2+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
−x2x2+1−1x2+1+(xx2+1+1)2x+x2+1x+x2+1- \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
3x(x2x2+1−1)(x2+1)32+3(xx2+1+1)(x2x2+1−1)(x+x2+1)x2+1+2(xx2+1+1)3(x+x2+1)2x+x2+1\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
The graph
Derivative of ln(x+sqrt(1+x^2))
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