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Identical expressions
(one +lnx)/(x^(four / fourteen)+ one)
(1 plus lnx) divide by (x to the power of (4 divide by 14) plus 1)
(one plus lnx) divide by (x to the power of (four divide by fourteen) plus one)
(1+lnx)/(x(4/14)+1)
1+lnx/x4/14+1
1+lnx/x^4/14+1
(1+lnx) divide by (x^(4 divide by 14)+1)
Similar expressions
(1+lnx)/(x^(4/14)-1)
(1-lnx)/(x^(4/14)+1)

The derivative of the function/ (1+lnx)/(x^(4/14)+1)

Derivative of (1+lnx)/(x^(4/14)+1)

The solution

You have entered [src]

1 + log(x)
----------
  2/7     
 x    + 1

$$\frac{\log{\left(x \right)} + 1}{x^{\frac{2}{7}} + 1}$$

(1 + log(x))/(x^(2/7) + 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x \right)} + 1$ and $g{\left(x \right)} = x^{\frac{2}{7}} + 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{1}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{\frac{2}{7}} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{\frac{2}{7}}$ goes to $\frac{2}{7 x^{\frac{5}{7}}}$
  The result is: $\frac{2}{7 x^{\frac{5}{7}}}$
Now plug in to the quotient rule:

$\frac{\frac{x^{\frac{2}{7}} + 1}{x} - \frac{2 \left(\log{\left(x \right)} + 1\right)}{7 x^{\frac{5}{7}}}}{\left(x^{\frac{2}{7}} + 1\right)^{2}}$
Now simplify:

$\frac{x^{\frac{5}{7}} - \frac{2 x \log{\left(x \right)}}{7} + \frac{5 x}{7}}{x^{\frac{16}{7}} + x^{\frac{12}{7}} + 2 x^{2}}$

The answer is:

$\frac{x^{\frac{5}{7}} - \frac{2 x \log{\left(x \right)}}{7} + \frac{5 x}{7}}{x^{\frac{16}{7}} + x^{\frac{12}{7}} + 2 x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     1           2*(1 + log(x))  
------------ - ------------------
  / 2/7    \                    2
x*\x    + 1/      5/7 / 2/7    \ 
               7*x   *\x    + 1/

$$\frac{1}{x \left(x^{\frac{2}{7}} + 1\right)} - \frac{2 \left(\log{\left(x \right)} + 1\right)}{7 x^{\frac{5}{7}} \left(x^{\frac{2}{7}} + 1\right)^{2}}$$

Simplify

The second derivative [src]

                                           /   4        5  \
                            2*(1 + log(x))*|-------- + ----|
                                           |     2/7    2/7|
  1            4                           \1 + x      x   /
- -- - ------------------ + --------------------------------
   2      12/7 /     2/7\             10/7 /     2/7\       
  x    7*x    *\1 + x   /         49*x    *\1 + x   /       
------------------------------------------------------------
                               2/7                          
                          1 + x

$$\frac{- \frac{1}{x^{2}} + \frac{2 \left(\frac{4}{x^{\frac{2}{7}} + 1} + \frac{5}{x^{\frac{2}{7}}}\right) \left(\log{\left(x \right)} + 1\right)}{49 x^{\frac{10}{7}} \left(x^{\frac{2}{7}} + 1\right)} - \frac{4}{7 x^{\frac{12}{7}} \left(x^{\frac{2}{7}} + 1\right)}}{x^{\frac{2}{7}} + 1}$$

Simplify

The third derivative [src]

  /                                                                /     2         5            5       \\
  |                            /   4        5  \   12*(1 + log(x))*|----------- + ---- + ---------------||
  |                          3*|-------- + ----|                   |          2    4/7    2/7 /     2/7\||
  |                            |     2/7    2/7|                   |/     2/7\    x      x   *\1 + x   /||
  |1            3              \1 + x      x   /                   \\1 + x   /                          /|
2*|-- + ------------------ + ------------------- - ------------------------------------------------------|
  | 3      19/7 /     2/7\       17/7 /     2/7\                         15/7 /     2/7\                 |
  \x    7*x    *\1 + x   /   49*x    *\1 + x   /                    343*x    *\1 + x   /                 /
----------------------------------------------------------------------------------------------------------
                                                      2/7                                                 
                                                 1 + x

$$\frac{2 \left(\frac{1}{x^{3}} - \frac{12 \left(\log{\left(x \right)} + 1\right) \left(\frac{2}{\left(x^{\frac{2}{7}} + 1\right)^{2}} + \frac{5}{x^{\frac{2}{7}} \left(x^{\frac{2}{7}} + 1\right)} + \frac{5}{x^{\frac{4}{7}}}\right)}{343 x^{\frac{15}{7}} \left(x^{\frac{2}{7}} + 1\right)} + \frac{3 \left(\frac{4}{x^{\frac{2}{7}} + 1} + \frac{5}{x^{\frac{2}{7}}}\right)}{49 x^{\frac{17}{7}} \left(x^{\frac{2}{7}} + 1\right)} + \frac{3}{7 x^{\frac{19}{7}} \left(x^{\frac{2}{7}} + 1\right)}\right)}{x^{\frac{2}{7}} + 1}$$

Simplify

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Derivative of (1+lnx)/(x^(4/14)+1)

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The solution