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Identical expressions
(((x^ three)/(one -x))-(sqrt(x)))*(ln((x^ two)+ one))
(((x cubed ) divide by (1 minus x)) minus ( square root of (x))) multiply by (ln((x squared ) plus 1))
(((x to the power of three) divide by (one minus x)) minus ( square root of (x))) multiply by (ln((x to the power of two) plus one))
(((x^3)/(1-x))-(√(x)))*(ln((x^2)+1))
(((x3)/(1-x))-(sqrt(x)))*(ln((x2)+1))
x3/1-x-sqrtx*lnx2+1
(((x³)/(1-x))-(sqrt(x)))*(ln((x²)+1))
(((x to the power of 3)/(1-x))-(sqrt(x)))*(ln((x to the power of 2)+1))
(((x^3)/(1-x))-(sqrt(x)))(ln((x^2)+1))
(((x3)/(1-x))-(sqrt(x)))(ln((x2)+1))
x3/1-x-sqrtxlnx2+1
x^3/1-x-sqrtxlnx^2+1
(((x^3) divide by (1-x))-(sqrt(x)))*(ln((x^2)+1))
Similar expressions
(((x^3)/(1-x))+(sqrt(x)))*(ln((x^2)+1))
(((x^3)/(1-x))-(sqrt(x)))*(ln((x^2)-1))
(((x^3)/(1+x))-(sqrt(x)))*(ln((x^2)+1))

The derivative of the function/ (((x^3)/(1-x))-(sqrt(x)))*(ln((x^2)+1))

Derivative of (((x^3)/(1-x))-(sqrt(x)))*(ln((x^2)+1))

The solution

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/   3         \            
|  x       ___|    / 2    \
|----- - \/ x |*log\x  + 1/
\1 - x        /

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

(x^3/(1 - x) - sqrt(x))*log(x^2 + 1)

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  /               2           3           \                   /              3         2 \     /         2 \ /           3  \
  |   1       24*x         8*x       24*x |    /     2\       |  1        2*x       6*x  |     |      2*x  | |  ___     x   |
  |- ---- - --------- + --------- + ------|*log\1 + x /   2*x*|----- - --------- + ------|   2*|-1 + ------|*|\/ x  + ------|
  |   3/2           2           3   -1 + x|                   |  ___           2   -1 + x|     |          2| \        -1 + x/
  \  x      (-1 + x)    (-1 + x)          /                   \\/ x    (-1 + x)          /     \     1 + x /                 
- ----------------------------------------------------- - -------------------------------- + --------------------------------
                            4                                               2                                  2             
                                                                       1 + x                              1 + x

$$- \frac{2 x \left(- \frac{2 x^{3}}{\left(x - 1\right)^{2}} + \frac{6 x^{2}}{x - 1} + \frac{1}{\sqrt{x}}\right)}{x^{2} + 1} + \frac{2 \left(\sqrt{x} + \frac{x^{3}}{x - 1}\right) \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1} - \frac{\left(\frac{8 x^{3}}{\left(x - 1\right)^{3}} - \frac{24 x^{2}}{\left(x - 1\right)^{2}} + \frac{24 x}{x - 1} - \frac{1}{x^{\frac{3}{2}}}\right) \log{\left(x^{2} + 1 \right)}}{4}$$

Simplify

The third derivative [src]

    /                                  3           2  \                 /         2 \ /              3         2 \       /               2           3           \       /         2 \ /           3  \
    | 1       16        48*x       16*x        48*x   |    /     2\     |      2*x  | |  1        2*x       6*x  |       |   1       24*x         8*x       24*x |       |      4*x  | |  ___     x   |
  3*|---- + ------ - --------- - --------- + ---------|*log\1 + x /   3*|-1 + ------|*|----- - --------- + ------|   3*x*|- ---- - --------- + --------- + ------|   4*x*|-3 + ------|*|\/ x  + ------|
    | 5/2   -1 + x           2           4           3|                 |          2| |  ___           2   -1 + x|       |   3/2           2           3   -1 + x|       |          2| \        -1 + x/
    \x               (-1 + x)    (-1 + x)    (-1 + x) /                 \     1 + x / \\/ x    (-1 + x)          /       \  x      (-1 + x)    (-1 + x)          /       \     1 + x /                 
- ----------------------------------------------------------------- + -------------------------------------------- - --------------------------------------------- - ----------------------------------
                                  8                                                           2                                          /     2\                                        2             
                                                                                         1 + x                                         2*\1 + x /                                /     2\              
                                                                                                                                                                                 \1 + x /

$$- \frac{4 x \left(\sqrt{x} + \frac{x^{3}}{x - 1}\right) \left(\frac{4 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{2}} - \frac{3 x \left(\frac{8 x^{3}}{\left(x - 1\right)^{3}} - \frac{24 x^{2}}{\left(x - 1\right)^{2}} + \frac{24 x}{x - 1} - \frac{1}{x^{\frac{3}{2}}}\right)}{2 \left(x^{2} + 1\right)} - \frac{3 \left(- \frac{16 x^{3}}{\left(x - 1\right)^{4}} + \frac{48 x^{2}}{\left(x - 1\right)^{3}} - \frac{48 x}{\left(x - 1\right)^{2}} + \frac{16}{x - 1} + \frac{1}{x^{\frac{5}{2}}}\right) \log{\left(x^{2} + 1 \right)}}{8} + \frac{3 \left(\frac{2 x^{2}}{x^{2} + 1} - 1\right) \left(- \frac{2 x^{3}}{\left(x - 1\right)^{2}} + \frac{6 x^{2}}{x - 1} + \frac{1}{\sqrt{x}}\right)}{x^{2} + 1}$$

Simplify

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Derivative of (((x^3)/(1-x))-(sqrt(x)))*(ln((x^2)+1))

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The solution