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Identical expressions
log((x+ three)/(x- three))
logarithm of ((x plus 3) divide by (x minus 3))
logarithm of ((x plus three) divide by (x minus three))
logx+3/x-3
log((x+3) divide by (x-3))
Similar expressions
log((x+3)/(x+3))
log((x-3)/(x-3))

The derivative of the function/ log((x+3)/(x-3))

Derivative of log((x+3)/(x-3))

The solution

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   /x + 3\
log|-----|
   \x - 3/

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

log((x + 3)/(x - 3))

Detail solution

Let $u = \frac{x + 3}{x - 3}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x + 3}{x - 3}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x + 3$ and $g{\left(x \right)} = x - 3$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $x + 3$ term by term:
    1. The derivative of the constant $3$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 3$ term by term:
    1. The derivative of the constant $-3$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $- \frac{6}{\left(x - 3\right)^{2}}$
The result of the chain rule is:

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

$- \frac{6}{x^{2} - 9}$

The answer is:

$- \frac{6}{x^{2} - 9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of log((x+3)/(x-3))

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The solution