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

Function f() - derivative -N order at the point
v

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The solution

You have entered [src]
   x            
-------*log(3*x)
2*x - 1         
x2x1log(3x)\frac{x}{2 x - 1} \log{\left(3 x \right)}
(x/(2*x - 1))*log(3*x)
Detail solution
  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=xlog(3x)f{\left(x \right)} = x \log{\left(3 x \right)} and g(x)=2x1g{\left(x \right)} = 2 x - 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the product rule:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: xx goes to 11

      g(x)=log(3x)g{\left(x \right)} = \log{\left(3 x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Let u=3xu = 3 x.

      2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

      3. Then, apply the chain rule. Multiply by ddx3x\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: xx goes to 11

          So, the result is: 33

        The result of the chain rule is:

        1x\frac{1}{x}

      The result is: log(3x)+1\log{\left(3 x \right)} + 1

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate 2x12 x - 1 term by term:

      1. The derivative of the constant 1-1 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: xx goes to 11

        So, the result is: 22

      The result is: 22

    Now plug in to the quotient rule:

    2xlog(3x)+(2x1)(log(3x)+1)(2x1)2\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}}

  2. Now simplify:

    2xlog(x)log(3)14x24x+1\frac{2 x - \log{\left(x \right)} - \log{\left(3 \right)} - 1}{4 x^{2} - 4 x + 1}


The answer is:

2xlog(x)log(3)14x24x+1\frac{2 x - \log{\left(x \right)} - \log{\left(3 \right)} - 1}{4 x^{2} - 4 x + 1}

The graph
02468-8-6-4-2-1010-50005000
The first derivative [src]
   1      /   1         2*x    \         
------- + |------- - ----------|*log(3*x)
2*x - 1   |2*x - 1            2|         
          \          (2*x - 1) /         
(2x(2x1)2+12x1)log(3x)+12x1\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}
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                      
4(2x2x11)log(3x)2x12(2x2x11)x1x2x1\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}
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                                
24(2x2x11)log(3x)(2x1)2+12(2x2x11)x(2x1)+3(2x2x11)x2+2x22x1\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}