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Similar expressions
cbrt(x)/(1-ln^2(x))

The derivative of the function/ cbrt(x)/(1+ln^2(x))

Derivative of cbrt(x)/(1+ln^2(x))

The solution

You have entered [src]

   3 ___   
   \/ x    
-----------
       2   
1 + log (x)

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

x^(1/3)/(1 + log(x)^2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt[3]{x}$ and $g{\left(x \right)} = \log{\left(x \right)}^{2} + 1$ .

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

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

$\frac{\log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 1}{3 x^{\frac{2}{3}} \left(\log{\left(x \right)}^{2} + 1\right)^{2}}$

The answer is:

$\frac{\log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 1}{3 x^{\frac{2}{3}} \left(\log{\left(x \right)}^{2} + 1\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         1                   2*log(x)     
-------------------- - -------------------
   2/3 /       2   \                     2
3*x   *\1 + log (x)/    2/3 /       2   \ 
                       x   *\1 + log (x)/

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

Simplify

The second derivative [src]

  /                 2                               \
  |            4*log (x)                            |
  |      -1 + ----------- + log(x)                  |
  |                  2                              |
  |  1        1 + log (x)                2*log(x)   |
2*|- - + ------------------------- - ---------------|
  |  9                 2               /       2   \|
  \             1 + log (x)          3*\1 + log (x)//
-----------------------------------------------------
                   5/3 /       2   \                 
                  x   *\1 + log (x)/

$$\frac{2 \left(- \frac{1}{9} + \frac{\log{\left(x \right)} - 1 + \frac{4 \log{\left(x \right)}^{2}}{\log{\left(x \right)}^{2} + 1}}{\log{\left(x \right)}^{2} + 1} - \frac{2 \log{\left(x \right)}}{3 \left(\log{\left(x \right)}^{2} + 1\right)}\right)}{x^{\frac{5}{3}} \left(\log{\left(x \right)}^{2} + 1\right)}$$

Simplify

The third derivative [src]

  /                                                                      2              3                       \
  |                2                                 12*log(x)     12*log (x)     24*log (x)                    |
  |           4*log (x)             -3 + 2*log(x) - ----------- + ----------- + --------------                  |
  |     -1 + ----------- + log(x)                          2             2                   2                  |
  |                 2                               1 + log (x)   1 + log (x)   /       2   \                   |
  |5         1 + log (x)                                                        \1 + log (x)/        2*log(x)   |
2*|-- + ------------------------- - ---------------------------------------------------------- + ---------------|
  |27                 2                                           2                                /       2   \|
  \            1 + log (x)                                 1 + log (x)                           3*\1 + log (x)//
-----------------------------------------------------------------------------------------------------------------
                                                 8/3 /       2   \                                               
                                                x   *\1 + log (x)/

$$\frac{2 \left(\frac{5}{27} + \frac{\log{\left(x \right)} - 1 + \frac{4 \log{\left(x \right)}^{2}}{\log{\left(x \right)}^{2} + 1}}{\log{\left(x \right)}^{2} + 1} - \frac{2 \log{\left(x \right)} - 3 + \frac{12 \log{\left(x \right)}^{2}}{\log{\left(x \right)}^{2} + 1} - \frac{12 \log{\left(x \right)}}{\log{\left(x \right)}^{2} + 1} + \frac{24 \log{\left(x \right)}^{3}}{\left(\log{\left(x \right)}^{2} + 1\right)^{2}}}{\log{\left(x \right)}^{2} + 1} + \frac{2 \log{\left(x \right)}}{3 \left(\log{\left(x \right)}^{2} + 1\right)}\right)}{x^{\frac{8}{3}} \left(\log{\left(x \right)}^{2} + 1\right)}$$

Simplify

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Derivative of cbrt(x)/(1+ln^2(x))

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The solution