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Identical expressions
ln((two -(x)^ three)/ three *x)
ln((2 minus (x) cubed ) divide by 3 multiply by x)
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ln((2-(x)3)/3*x)
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ln((2-(x) to the power of 3)/3*x)
ln((2-(x)^3)/3x)
ln((2-(x)3)/3x)
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ln((2-(x)^3) divide by 3*x)
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ln((2+(x)^3)/3*x)

The derivative of the function/ ln((2-(x)^3)/3*x)

Derivative of ln((2-(x)^3)/3*x)

The solution

You have entered [src]

   /     3  \
   |2 - x   |
log|------*x|
   \  3     /

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

log(((2 - x^3)/3)*x)

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

  /            3\
  |   3   2 - x |
3*|- x  + ------|
  \         3   /
-----------------
      /     3\   
    x*\2 - x /

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

Simplify

The second derivative [src]

  /              3       /        3\\
  |      -1 + 2*x    3*x*\-1 + 2*x /|
2*|6*x - --------- - ---------------|
  |           2                3    |
  \          x           -2 + x     /
-------------------------------------
                     3               
               -2 + x

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

Simplify

The third derivative [src]

  /        3        3       3 /        3\\
  |-1 + 2*x     18*x     9*x *\-1 + 2*x /|
4*|--------- - ------- + ----------------|
  |     3            3               2   |
  |    x       -2 + x       /      3\    |
  \                         \-2 + x /    /
------------------------------------------
                       3                  
                 -2 + x

$$\frac{4 \left(- \frac{18 x^{3}}{x^{3} - 2} + \frac{9 x^{3} \left(2 x^{3} - 1\right)}{\left(x^{3} - 2\right)^{2}} + \frac{2 x^{3} - 1}{x^{3}}\right)}{x^{3} - 2}$$

Simplify

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Derivative of ln((2-(x)^3)/3*x)

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The solution