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Identical expressions
(ln(x))/(sqrt(x^ two - one))
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Similar expressions
(ln(x))/(sqrt(x^2+1))

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

Derivative of (ln(x))/(sqrt(x^2-1))

The solution

You have entered [src]

   log(x)  
-----------
   ________
  /  2     
\/  x  - 1

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

log(x)/sqrt(x^2 - 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = \sqrt{x^{2} - 1}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1           x*log(x) 
------------- - -----------
     ________           3/2
    /  2        / 2    \   
x*\/  x  - 1    \x  - 1/

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

Simplify

The second derivative [src]

                 /          2 \       
                 |       3*x  |       
                 |-1 + -------|*log(x)
                 |           2|       
  1       2      \     -1 + x /       
- -- - ------- + ---------------------
   2         2                2       
  x    -1 + x           -1 + x        
--------------------------------------
                _________             
               /       2              
             \/  -1 + x

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

Simplify

The third derivative [src]

                     /          2 \       /          2 \       
                     |       3*x  |       |       5*x  |       
                   3*|-1 + -------|   3*x*|-3 + -------|*log(x)
                     |           2|       |           2|       
2         3          \     -1 + x /       \     -1 + x /       
-- + ----------- + ---------------- - -------------------------
 3     /      2\       /      2\                       2       
x    x*\-1 + x /     x*\-1 + x /              /      2\        
                                              \-1 + x /        
---------------------------------------------------------------
                             _________                         
                            /       2                          
                          \/  -1 + x

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

Simplify

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Derivative of (ln(x))/(sqrt(x^2-1))

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The solution