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

Derivative of x/(2*x-1)

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

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The solution

You have entered [src] 
   x   
-------
2*x - 1
x2x−1\frac{x}{2 x - 1} 
d /   x   \
--|-------|
dx\2*x - 1/
ddxx2x−1\frac{d}{d x} \frac{x}{2 x - 1}
Transform
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)=xf{\left(x \right)} = x and g(x)=2x−1g{\left(x \right)} = 2 x - 1.

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

    1. Apply the power rule: xx goes to 11

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

    1. Differentiate 2x−12 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:

    −1(2x−1)2- \frac{1}{\left(2 x - 1\right)^{2}}

The answer is:

−1(2x−1)2- \frac{1}{\left(2 x - 1\right)^{2}}

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   1         2*x    
------- - ----------
2*x - 1            2
          (2*x - 1)
−2x(2x−1)2+12x−1- \frac{2 x}{\left(2 x - 1\right)^{2}} + \frac{1}{2 x - 1}
Simplify
The second derivative [src] 
  /       2*x   \
4*|-1 + --------|
  \     -1 + 2*x/
-----------------
             2   
   (-1 + 2*x)
4⋅(2x2x−1−1)(2x−1)2\frac{4 \cdot \left(\frac{2 x}{2 x - 1} - 1\right)}{\left(2 x - 1\right)^{2}}
Simplify
The third derivative [src] 
   /      2*x   \
24*|1 - --------|
   \    -1 + 2*x/
-----------------
             3   
   (-1 + 2*x)
24(−2x2x−1+1)(2x−1)3\frac{24 \left(- \frac{2 x}{2 x - 1} + 1\right)}{\left(2 x - 1\right)^{3}}
Simplify
The graph
Derivative of x/(2*x-1)
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