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Derivative of ((2x+3)/(x-1))^(1/4)

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

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The solution

You have entered [src]
    _________
   / 2*x + 3 
4 /  ------- 
\/    x - 1  
2x+3x14\sqrt[4]{\frac{2 x + 3}{x - 1}}
((2*x + 3)/(x - 1))^(1/4)
Detail solution
  1. Let u=2x+3x1u = \frac{2 x + 3}{x - 1}.

  2. Apply the power rule: u4\sqrt[4]{u} goes to 14u34\frac{1}{4 u^{\frac{3}{4}}}

  3. Then, apply the chain rule. Multiply by ddx2x+3x1\frac{d}{d x} \frac{2 x + 3}{x - 1}:

    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)=2x+3f{\left(x \right)} = 2 x + 3 and g(x)=x1g{\left(x \right)} = x - 1.

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

      1. Differentiate 2x+32 x + 3 term by term:

        1. The derivative of the constant 33 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

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

      1. Differentiate x1x - 1 term by term:

        1. The derivative of the constant 1-1 is zero.

        2. Apply the power rule: xx goes to 11

        The result is: 11

      Now plug in to the quotient rule:

      5(x1)2- \frac{5}{\left(x - 1\right)^{2}}

    The result of the chain rule is:

    54(2x+3x1)34(x1)2- \frac{5}{4 \left(\frac{2 x + 3}{x - 1}\right)^{\frac{3}{4}} \left(x - 1\right)^{2}}

  4. Now simplify:

    54(2x+3x1)34(x1)2- \frac{5}{4 \left(\frac{2 x + 3}{x - 1}\right)^{\frac{3}{4}} \left(x - 1\right)^{2}}


The answer is:

54(2x+3x1)34(x1)2- \frac{5}{4 \left(\frac{2 x + 3}{x - 1}\right)^{\frac{3}{4}} \left(x - 1\right)^{2}}

The graph
02468-8-6-4-2-1010-1010
The first derivative [src]
    _________                                 
   / 2*x + 3          /    1        2*x + 3  \
4 /  ------- *(x - 1)*|--------- - ----------|
\/    x - 1           |2*(x - 1)            2|
                      \            4*(x - 1) /
----------------------------------------------
                   2*x + 3                    
2x+3x14(x1)(12(x1)2x+34(x1)2)2x+3\frac{\sqrt[4]{\frac{2 x + 3}{x - 1}} \left(x - 1\right) \left(\frac{1}{2 \left(x - 1\right)} - \frac{2 x + 3}{4 \left(x - 1\right)^{2}}\right)}{2 x + 3}
The second derivative [src]
                            /                         3 + 2*x\
    _________               |                     2 - -------|
   / 3 + 2*x  /    3 + 2*x\ |     8        4           -1 + x|
4 /  ------- *|2 - -------|*|- ------- - ------ + -----------|
\/    -1 + x  \     -1 + x/ \  3 + 2*x   -1 + x     3 + 2*x  /
--------------------------------------------------------------
                         16*(3 + 2*x)                         
2x+3x14(22x+3x1)(22x+3x12x+382x+34x1)16(2x+3)\frac{\sqrt[4]{\frac{2 x + 3}{x - 1}} \left(2 - \frac{2 x + 3}{x - 1}\right) \left(\frac{2 - \frac{2 x + 3}{x - 1}}{2 x + 3} - \frac{8}{2 x + 3} - \frac{4}{x - 1}\right)}{16 \left(2 x + 3\right)}
The third derivative [src]
                            /                                                                               2                        \
                            |                                                  /    3 + 2*x\   /    3 + 2*x\         /    3 + 2*x\   |
    _________               |                                                3*|2 - -------|   |2 - -------|       3*|2 - -------|   |
   / 3 + 2*x  /    3 + 2*x\ |     1            2                1              \     -1 + x/   \     -1 + x/         \     -1 + x/   |
4 /  ------- *|2 - -------|*|----------- + ---------- + ------------------ - --------------- + -------------- - ---------------------|
\/    -1 + x  \     -1 + x/ |          2            2   (-1 + x)*(3 + 2*x)                2                2    16*(-1 + x)*(3 + 2*x)|
                            \2*(-1 + x)    (3 + 2*x)                           8*(3 + 2*x)     64*(3 + 2*x)                          /
--------------------------------------------------------------------------------------------------------------------------------------
                                                               3 + 2*x                                                                
2x+3x14(22x+3x1)((22x+3x1)264(2x+3)23(22x+3x1)8(2x+3)23(22x+3x1)16(x1)(2x+3)+2(2x+3)2+1(x1)(2x+3)+12(x1)2)2x+3\frac{\sqrt[4]{\frac{2 x + 3}{x - 1}} \left(2 - \frac{2 x + 3}{x - 1}\right) \left(\frac{\left(2 - \frac{2 x + 3}{x - 1}\right)^{2}}{64 \left(2 x + 3\right)^{2}} - \frac{3 \left(2 - \frac{2 x + 3}{x - 1}\right)}{8 \left(2 x + 3\right)^{2}} - \frac{3 \left(2 - \frac{2 x + 3}{x - 1}\right)}{16 \left(x - 1\right) \left(2 x + 3\right)} + \frac{2}{\left(2 x + 3\right)^{2}} + \frac{1}{\left(x - 1\right) \left(2 x + 3\right)} + \frac{1}{2 \left(x - 1\right)^{2}}\right)}{2 x + 3}