Derivative of sqrt(log(x))

The solution

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  ________
\/ log(x)

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

d /  ________\
--\\/ log(x) /
dx

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1       
--------------
      ________
2*x*\/ log(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of sqrt(log(x))

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The solution