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Identical expressions
ln^ two (sqrtx-x^ two)
ln squared ( square root of x minus x squared )
ln to the power of two ( square root of x minus x to the power of two)
ln^2(√x-x^2)
ln2(sqrtx-x2)
ln2sqrtx-x2
ln²(sqrtx-x²)
ln to the power of 2(sqrtx-x to the power of 2)
ln^2sqrtx-x^2
Similar expressions
ln^2(sqrtx+x^2)

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

Derivative of ln^2(sqrtx-x^2)

The solution

You have entered [src]

   2/  ___    2\
log \\/ x  - x /

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

log(sqrt(x) - x^2)^2

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

               2                                               2                
/    1        \                                 /    1        \     /  ___    2\
|- ----- + 4*x|                                 |- ----- + 4*x| *log\\/ x  - x /
|    ___      |                                 |    ___      |                 
\  \/ x       /    /     1  \    /  ___    2\   \  \/ x       /                 
---------------- - |8 + ----|*log\\/ x  - x / - --------------------------------
     ___    2      |     3/2|                                ___    2           
   \/ x  - x       \    x   /                              \/ x  - x            
--------------------------------------------------------------------------------
                                   /  ___    2\                                 
                                 2*\\/ x  - x /

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

Simplify

The third derivative [src]

                                     3                    3                                                                                              
                      /    1        \      /    1        \     /  ___    2\     /     1  \ /    1        \     /     1  \ /    1        \    /  ___    2\
                    3*|- ----- + 4*x|    2*|- ----- + 4*x| *log\\/ x  - x /   3*|8 + ----|*|- ----- + 4*x|   3*|8 + ----|*|- ----- + 4*x|*log\\/ x  - x /
     /  ___    2\     |    ___      |      |    ___      |                      |     3/2| |    ___      |     |     3/2| |    ___      |                
3*log\\/ x  - x /     \  \/ x       /      \  \/ x       /                      \    x   / \  \/ x       /     \    x   / \  \/ x       /                
----------------- + ------------------ - ---------------------------------- + ---------------------------- - --------------------------------------------
        5/2                       2                            2                         ___    2                               ___    2                 
       x              /  ___    2\                 /  ___    2\                        \/ x  - x                              \/ x  - x                  
                      \\/ x  - x /                 \\/ x  - x /                                                                                          
---------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                        /  ___    2\                                                                     
                                                                      4*\\/ x  - x /

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

Simplify

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Derivative of ln^2(sqrtx-x^2)

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The solution