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Identical expressions
ln((3x)/(x^ two - one))
ln((3x) divide by (x squared minus 1))
ln((3x) divide by (x to the power of two minus one))
ln((3x)/(x2-1))
ln3x/x2-1
ln((3x)/(x²-1))
ln((3x)/(x to the power of 2-1))
ln3x/x^2-1
ln((3x) divide by (x^2-1))
Similar expressions
ln((3x)/(x^2+1))

The derivative of the function/ ln((3x)/(x^2-1))

Derivative of ln((3x)/(x^2-1))

The solution

You have entered [src]

   / 3*x  \
log|------|
   | 2    |
   \x  - 1/

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

log((3*x)/(x^2 - 1))

Detail solution

Let $u = \frac{3 x}{x^{2} - 1}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{3 x}{x^{2} - 1}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 3 x$ and $g{\left(x \right)} = x^{2} - 1$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $3$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  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: $2 x$
  Now plug in to the quotient rule:
  
  $\frac{- 3 x^{2} - 3}{\left(x^{2} - 1\right)^{2}}$
The result of the chain rule is:

$\frac{\left(- 3 x^{2} - 3\right) \left(x^{2} - 1\right)}{3 x \left(x^{2} - 1\right)^{2}}$
Now simplify:

$- \frac{x^{2} + 1}{x^{3} - x}$

The answer is:

$- \frac{x^{2} + 1}{x^{3} - x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         /               2  \
/ 2    \ |  3         6*x   |
\x  - 1/*|------ - ---------|
         | 2               2|
         |x  - 1   / 2    \ |
         \         \x  - 1/ /
-----------------------------
             3*x

$$\frac{\left(x^{2} - 1\right) \left(- \frac{6 x^{2}}{\left(x^{2} - 1\right)^{2}} + \frac{3}{x^{2} - 1}\right)}{3 x}$$

Simplify

The second derivative [src]

                    2                   /          2 \
                 2*x                    |       4*x  |
          -1 + -------                2*|-3 + -------|
                     2         2        |           2|
   2           -1 + x       4*x         \     -1 + x /
------- + ------------ - ---------- + ----------------
      2         2                 2             2     
-1 + x         x         /      2\        -1 + x      
                         \-1 + x /

$$- \frac{4 x^{2}}{\left(x^{2} - 1\right)^{2}} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 3\right)}{x^{2} - 1} + \frac{2}{x^{2} - 1} + \frac{\frac{2 x^{2}}{x^{2} - 1} - 1}{x^{2}}$$

Simplify

The third derivative [src]

  /                                                            /         2          4   \                   \
  |            2                                        2      |      8*x        8*x    |     /          2 \|
  |         2*x                                      2*x     3*|1 - ------- + ----------|     |       4*x  ||
  |  -1 + -------                             -1 + -------     |          2            2|   2*|-3 + -------||
  |             2                      3                 2     |    -1 + x    /      2\ |     |           2||
  |       -1 + x       12*x        16*x            -1 + x      \              \-1 + x / /     \     -1 + x /|
2*|- ------------ - ---------- + ---------- + ------------ - ---------------------------- - ----------------|
  |        3                 2            3     /      2\              /      2\                /      2\   |
  |       x         /      2\    /      2\    x*\-1 + x /            x*\-1 + x /              x*\-1 + x /   |
  \                 \-1 + x /    \-1 + x /                                                                  /

$$2 \left(\frac{16 x^{3}}{\left(x^{2} - 1\right)^{3}} - \frac{12 x}{\left(x^{2} - 1\right)^{2}} + \frac{\frac{2 x^{2}}{x^{2} - 1} - 1}{x \left(x^{2} - 1\right)} - \frac{2 \left(\frac{4 x^{2}}{x^{2} - 1} - 3\right)}{x \left(x^{2} - 1\right)} - \frac{3 \left(\frac{8 x^{4}}{\left(x^{2} - 1\right)^{2}} - \frac{8 x^{2}}{x^{2} - 1} + 1\right)}{x \left(x^{2} - 1\right)} - \frac{\frac{2 x^{2}}{x^{2} - 1} - 1}{x^{3}}\right)$$

Simplify

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Derivative of ln((3x)/(x^2-1))

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The solution