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Identical expressions
(x/(2x- one))*ln(3x)
(x divide by (2x minus 1)) multiply by ln(3x)
(x divide by (2x minus one)) multiply by ln(3x)
(x/(2x-1))ln(3x)
x/2x-1ln3x
(x divide by (2x-1))*ln(3x)
Similar expressions
(x/(2x+1))*ln(3x)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x \log{\left(3 x \right)}$ and $g{\left(x \right)} = 2 x - 1$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

        /       2*x   \     /       2*x   \         
      2*|-1 + --------|   4*|-1 + --------|*log(3*x)
  1     \     -1 + 2*x/     \     -1 + 2*x/         
- - - ----------------- + --------------------------
  x           x                    -1 + 2*x         
----------------------------------------------------
                      -1 + 2*x

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

Simplify

The third derivative [src]

       /       2*x   \      /       2*x   \               /       2*x   \
     3*|-1 + --------|   24*|-1 + --------|*log(3*x)   12*|-1 + --------|
2      \     -1 + 2*x/      \     -1 + 2*x/               \     -1 + 2*x/
-- + ----------------- - --------------------------- + ------------------
 2            2                            2              x*(-1 + 2*x)   
x            x                   (-1 + 2*x)                              
-------------------------------------------------------------------------
                                 -1 + 2*x

$$\frac{- \frac{24 \left(\frac{2 x}{2 x - 1} - 1\right) \log{\left(3 x \right)}}{\left(2 x - 1\right)^{2}} + \frac{12 \left(\frac{2 x}{2 x - 1} - 1\right)}{x \left(2 x - 1\right)} + \frac{3 \left(\frac{2 x}{2 x - 1} - 1\right)}{x^{2}} + \frac{2}{x^{2}}}{2 x - 1}$$

Simplify

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

The graph:

Piecewise:

The solution