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Derivative of:
Derivative of x^12
Derivative of -e^x
Derivative of e^(2-x)
Derivative of x^2-4*x
Identical expressions
(ln^ three (x))^ one / two
(ln cubed (x)) to the power of 1 divide by 2
(ln to the power of three (x)) to the power of one divide by two
(ln3(x))1/2
ln3x1/2
(ln³(x))^1/2
(ln to the power of 3(x)) to the power of 1/2
ln^3x^1/2
(ln^3(x))^1 divide by 2

The derivative of the function/ (ln^3(x))^1/2

Derivative of (ln^3(x))^1/2

The solution

You have entered [src]

   _________
  /    3    
\/  log (x)

$$\sqrt{\log{\left(x \right)}^{3}}$$

  /   _________\
d |  /    3    |
--\\/  log (x) /
dx

$$\frac{d}{d x} \sqrt{\log{\left(x \right)}^{3}}$$

Transform

Detail solution

Let $u = \log{\left(x \right)}^{3}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}^{3}$ :
1. Let $u = \log{\left(x \right)}$ .
2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
3. 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:
  
  $\frac{3 \log{\left(x \right)}^{2}}{x}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

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The first derivative [src]

     _________
    /    3    
3*\/  log (x) 
--------------
  2*x*log(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     _________                           
    /    3     /       3           1    \
3*\/  log (x) *|1 - -------- - ---------|
               |    4*log(x)        2   |
               \               8*log (x)/
-----------------------------------------
                 3                       
                x *log(x)

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

Simplify

The graph

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Identical expressions

Derivative of (ln^3(x))^1/2

The graph:

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The solution