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

Derivative of sqrt(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{\sqrt{x}}{2 x - 1} 
sqrt(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)=xf{\left(x \right)} = \sqrt{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: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

    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:

    −2x+2x−12x(2x−1)2\frac{- 2 \sqrt{x} + \frac{2 x - 1}{2 \sqrt{x}}}{\left(2 x - 1\right)^{2}}

  2. Now simplify:

    −x+12x(2x−1)2- \frac{x + \frac{1}{2}}{\sqrt{x} \left(2 x - 1\right)^{2}}

The answer is:

−x+12x(2x−1)2- \frac{x + \frac{1}{2}}{\sqrt{x} \left(2 x - 1\right)^{2}}

The graph
02468-8-6-4-2-1010-2000020000
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The first derivative [src] 
                         ___  
        1            2*\/ x   
----------------- - ----------
    ___                      2
2*\/ x *(2*x - 1)   (2*x - 1)
−2x(2x−1)2+12x(2x−1)- \frac{2 \sqrt{x}}{\left(2 x - 1\right)^{2}} + \frac{1}{2 \sqrt{x} \left(2 x - 1\right)}
Simplify
The second derivative [src] 
                                    ___  
    1             2             8*\/ x   
- ------ - ---------------- + -----------
     3/2     ___                        2
  4*x      \/ x *(-1 + 2*x)   (-1 + 2*x) 
-----------------------------------------
                 -1 + 2*x
8x(2x−1)2−2x(2x−1)−14x322x−1\frac{\frac{8 \sqrt{x}}{\left(2 x - 1\right)^{2}} - \frac{2}{\sqrt{x} \left(2 x - 1\right)} - \frac{1}{4 x^{\frac{3}{2}}}}{2 x - 1}
Simplify
The third derivative [src] 
  /                                    ___                     \
  |  1              1             16*\/ x             4        |
3*|------ + ----------------- - ----------- + -----------------|
  |   5/2      3/2                        3     ___           2|
  \8*x      2*x   *(-1 + 2*x)   (-1 + 2*x)    \/ x *(-1 + 2*x) /
----------------------------------------------------------------
                            -1 + 2*x
3(−16x(2x−1)3+4x(2x−1)2+12x32(2x−1)+18x52)2x−1\frac{3 \left(- \frac{16 \sqrt{x}}{\left(2 x - 1\right)^{3}} + \frac{4}{\sqrt{x} \left(2 x - 1\right)^{2}} + \frac{1}{2 x^{\frac{3}{2}} \left(2 x - 1\right)} + \frac{1}{8 x^{\frac{5}{2}}}\right)}{2 x - 1}
Simplify
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