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Identical expressions
five /((x+lnx)^ one / five)
5 divide by ((x plus lnx) to the power of 1 divide by 5)
five divide by ((x plus lnx) to the power of one divide by five)
5/((x+lnx)1/5)
5/x+lnx1/5
5/x+lnx^1/5
5 divide by ((x+lnx)^1 divide by 5)
Similar expressions
5/((x-lnx)^1/5)

The derivative of the function/ 5/((x+lnx)^1/5)

Derivative of 5/((x+lnx)^1/5)

The solution

You have entered [src]

      5       
--------------
5 ____________
\/ x + log(x)

$$\frac{5}{\sqrt[5]{x + \log{\left(x \right)}}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sqrt[5]{x + \log{\left(x \right)}}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt[5]{x + \log{\left(x \right)}}$ :
  1. Let $u = x + \log{\left(x \right)}$ .
  2. Apply the power rule: $\sqrt[5]{u}$ goes to $\frac{1}{5 u^{\frac{4}{5}}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + \log{\left(x \right)}\right)$ :
    1. Differentiate $x + \log{\left(x \right)}$ term by term:
      1. Apply the power rule: $x$ goes to $1$
      2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result is: $1 + \frac{1}{x}$
    The result of the chain rule is:
    
    $\frac{1 + \frac{1}{x}}{5 \left(x + \log{\left(x \right)}\right)^{\frac{4}{5}}}$
  The result of the chain rule is:
  
  $- \frac{1 + \frac{1}{x}}{5 \left(x + \log{\left(x \right)}\right)^{\frac{6}{5}}}$
So, the result is: $- \frac{1 + \frac{1}{x}}{\left(x + \log{\left(x \right)}\right)^{\frac{6}{5}}}$
Now simplify:

$- \frac{x + 1}{x \left(x + \log{\left(x \right)}\right)^{\frac{6}{5}}}$

The answer is:

$- \frac{x + 1}{x \left(x + \log{\left(x \right)}\right)^{\frac{6}{5}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     /1    1 \ 
  -5*|- + ---| 
     \5   5*x/ 
---------------
            6/5
(x + log(x))

$$- \frac{5 \left(\frac{1}{5} + \frac{1}{5 x}\right)}{\left(x + \log{\left(x \right)}\right)^{\frac{6}{5}}}$$

Simplify

The second derivative [src]

               2 
        /    1\  
      6*|1 + -|  
 5      \    x/  
 -- + ---------- 
  2   x + log(x) 
 x               
-----------------
              6/5
5*(x + log(x))

$$\frac{\frac{6 \left(1 + \frac{1}{x}\right)^{2}}{x + \log{\left(x \right)}} + \frac{5}{x^{2}}}{5 \left(x + \log{\left(x \right)}\right)^{\frac{6}{5}}}$$

Simplify

The third derivative [src]

   /                3                   \
   |         /    1\           /    1\  |
   |      33*|1 + -|        45*|1 + -|  |
   |25       \    x/           \    x/  |
-2*|-- + ------------- + ---------------|
   | 3               2    2             |
   \x    (x + log(x))    x *(x + log(x))/
-----------------------------------------
                           6/5           
            25*(x + log(x))

$$- \frac{2 \left(\frac{33 \left(1 + \frac{1}{x}\right)^{3}}{\left(x + \log{\left(x \right)}\right)^{2}} + \frac{45 \left(1 + \frac{1}{x}\right)}{x^{2} \left(x + \log{\left(x \right)}\right)} + \frac{25}{x^{3}}\right)}{25 \left(x + \log{\left(x \right)}\right)^{\frac{6}{5}}}$$

Simplify

The graph

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Derivative of 5/((x+lnx)^1/5)

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The solution