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

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
  3*x  
-------
2*x - 1
3x2x−1\frac{3 x}{2 x - 1} 
(3*x)/(2*x - 1)
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)=3xf{\left(x \right)} = 3 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. 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

    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:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-5000050000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   3         6*x    
------- - ----------
2*x - 1            2
          (2*x - 1)
−6x(2x−1)2+32x−1- \frac{6 x}{\left(2 x - 1\right)^{2}} + \frac{3}{2 x - 1}
Simplify
The second derivative [src] 
   /       2*x   \
12*|-1 + --------|
   \     -1 + 2*x/
------------------
             2    
   (-1 + 2*x)
12(2x2x−1−1)(2x−1)2\frac{12 \left(\frac{2 x}{2 x - 1} - 1\right)}{\left(2 x - 1\right)^{2}}
Simplify
The third derivative [src] 
   /      2*x   \
72*|1 - --------|
   \    -1 + 2*x/
-----------------
             3   
   (-1 + 2*x)
72(−2x2x−1+1)(2x−1)3\frac{72 \left(- \frac{2 x}{2 x - 1} + 1\right)}{\left(2 x - 1\right)^{3}}
Simplify
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