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Similar expressions
ln(x+sqrt(x^2+a^2))
ln(x-sqrt(x^2-a^2))

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

Derivative of ln(x+sqrt(x^2-a^2))

The solution

You have entered [src]

   /       _________\
   |      /  2    2 |
log\x + \/  x  - a  /

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

log(x + sqrt(x^2 - a^2))

Detail solution

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

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

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

The answer is:

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

The first derivative [src]

         x      
1 + ------------
       _________
      /  2    2 
    \/  x  - a  
----------------
       _________
      /  2    2 
x + \/  x  - a

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

Simplify

The second derivative [src]

 /                  2               \ 
 |/         x      \             2  | 
 ||1 + ------------|            x   | 
 ||       _________|    -1 + -------| 
 ||      /  2    2 |          2    2| 
 |\    \/  x  - a  /         x  - a | 
-|------------------- + ------------| 
 |         _________       _________| 
 |        /  2    2       /  2    2 | 
 \  x + \/  x  - a      \/  x  - a  / 
--------------------------------------
                  _________           
                 /  2    2            
           x + \/  x  - a

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

Simplify

The third derivative [src]

                    3                                             /         2  \
  /         x      \        /         2  \     /         x      \ |        x   |
2*|1 + ------------|        |        x   |   3*|1 + ------------|*|-1 + -------|
  |       _________|    3*x*|-1 + -------|     |       _________| |      2    2|
  |      /  2    2 |        |      2    2|     |      /  2    2 | \     x  - a /
  \    \/  x  - a  /        \     x  - a /     \    \/  x  - a  /               
--------------------- + ------------------ + -----------------------------------
                   2                3/2        /       _________\    _________  
 /       _________\        / 2    2\           |      /  2    2 |   /  2    2   
 |      /  2    2 |        \x  - a /           \x + \/  x  - a  /*\/  x  - a    
 \x + \/  x  - a  /                                                             
--------------------------------------------------------------------------------
                                       _________                                
                                      /  2    2                                 
                                x + \/  x  - a

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

Simplify

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Derivative of ln(x+sqrt(x^2-a^2))

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The solution