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Identical expressions
((2x+ three)/(x- one))^(one / four)
((2x plus 3) divide by (x minus 1)) to the power of (1 divide by 4)
((2x plus three) divide by (x minus one)) to the power of (one divide by four)
((2x+3)/(x-1))(1/4)
2x+3/x-11/4
2x+3/x-1^1/4
((2x+3) divide by (x-1))^(1 divide by 4)
Similar expressions
((2x+3)/(x+1))^(1/4)
((2x-3)/(x-1))^(1/4)

The derivative of the function/ ((2x+3)/(x-1))^(1/4)

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

The solution

You have entered [src]

    _________
   / 2*x + 3 
4 /  ------- 
\/    x - 1

$$\sqrt[4]{\frac{2 x + 3}{x - 1}}$$

((2*x + 3)/(x - 1))^(1/4)

Detail solution

Let $u = \frac{2 x + 3}{x - 1}$ .
Apply the power rule: $\sqrt[4]{u}$ goes to $\frac{1}{4 u^{\frac{3}{4}}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{2 x + 3}{x - 1}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2 x + 3$ and $g{\left(x \right)} = x - 1$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $2 x + 3$ term by term:
    1. The derivative of the constant $3$ 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: $x$ goes to $1$
      So, the result is: $2$
    The result is: $2$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $- \frac{5}{\left(x - 1\right)^{2}}$
The result of the chain rule is:

$- \frac{5}{4 \left(\frac{2 x + 3}{x - 1}\right)^{\frac{3}{4}} \left(x - 1\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

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

$$\frac{\sqrt[4]{\frac{2 x + 3}{x - 1}} \left(x - 1\right) \left(\frac{1}{2 \left(x - 1\right)} - \frac{2 x + 3}{4 \left(x - 1\right)^{2}}\right)}{2 x + 3}$$

Simplify

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)

$$\frac{\sqrt[4]{\frac{2 x + 3}{x - 1}} \left(2 - \frac{2 x + 3}{x - 1}\right) \left(\frac{2 - \frac{2 x + 3}{x - 1}}{2 x + 3} - \frac{8}{2 x + 3} - \frac{4}{x - 1}\right)}{16 \left(2 x + 3\right)}$$

Simplify

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

$$\frac{\sqrt[4]{\frac{2 x + 3}{x - 1}} \left(2 - \frac{2 x + 3}{x - 1}\right) \left(\frac{\left(2 - \frac{2 x + 3}{x - 1}\right)^{2}}{64 \left(2 x + 3\right)^{2}} - \frac{3 \left(2 - \frac{2 x + 3}{x - 1}\right)}{8 \left(2 x + 3\right)^{2}} - \frac{3 \left(2 - \frac{2 x + 3}{x - 1}\right)}{16 \left(x - 1\right) \left(2 x + 3\right)} + \frac{2}{\left(2 x + 3\right)^{2}} + \frac{1}{\left(x - 1\right) \left(2 x + 3\right)} + \frac{1}{2 \left(x - 1\right)^{2}}\right)}{2 x + 3}$$

Simplify

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

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The solution