Derivative of sqrt((ln(5x)))

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  __________
\/ log(5*x)

$$\sqrt{\log{\left(5 x \right)}}$$

sqrt(log(5*x))

Detail solution

Let $u = \log{\left(5 x \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(5 x \right)}$ :
1. Let $u = 5 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
  1. 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: $5$
  The result of the chain rule is:
  
  $\frac{1}{x}$
The result of the chain rule is:

$\frac{1}{2 x \sqrt{\log{\left(5 x \right)}}}$

The answer is:

$\frac{1}{2 x \sqrt{\log{\left(5 x \right)}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       1        
----------------
      __________
2*x*\/ log(5*x)

$$\frac{1}{2 x \sqrt{\log{\left(5 x \right)}}}$$

Simplify

The second derivative [src]

  /       1    \ 
 -|2 + --------| 
  \    log(5*x)/ 
-----------------
   2   __________
4*x *\/ log(5*x)

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

Simplify

The third derivative [src]

        3             3     
1 + ---------- + -----------
    4*log(5*x)        2     
                 8*log (5*x)
----------------------------
       3   __________       
      x *\/ log(5*x)

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

Simplify

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Derivative of sqrt((ln(5x)))

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The solution