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Similar expressions
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The derivative of the function/ xln(x+(1+x^2)^1/2)

Derivative of xln(x+(1+x^2)^1/2)

The solution

You have entered [src]

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

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

x*log(x + sqrt(1 + x^2))

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \log{\left(x + \sqrt{x^{2} + 1} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + \sqrt{x^{2} + 1}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. 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:
        
        The derivative of the constant $1$ is zero.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        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:
  
  $\frac{\frac{x}{\sqrt{x^{2} + 1}} + 1}{x + \sqrt{x^{2} + 1}}$
The result is: $\frac{x \left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)}{x + \sqrt{x^{2} + 1}} + \log{\left(x + \sqrt{x^{2} + 1} \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Similar expressions

Derivative of xln(x+(1+x^2)^1/2)

The graph:

Piecewise:

The solution