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Identical expressions
sqrt(one +ln(x))/x
square root of (1 plus ln(x)) divide by x
square root of (one plus ln(x)) divide by x
√(1+ln(x))/x
sqrt1+lnx/x
sqrt(1+ln(x)) divide by x
Similar expressions
sqrt(1-ln(x))/x

The derivative of the function/ sqrt(1+ln(x))/x

Derivative of sqrt(1+ln(x))/x

The solution

You have entered [src]

  ____________
\/ 1 + log(x) 
--------------
      x

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

sqrt(1 + log(x))/x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sqrt{\log{\left(x \right)} + 1}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(x \right)} + 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\log{\left(x \right)} + 1\right)$ :
  1. Differentiate $\log{\left(x \right)} + 1$ term by term:
    1. The derivative of the constant $1$ is zero.
    2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result is: $\frac{1}{x}$
  The result of the chain rule is:
  
  $\frac{1}{2 x \sqrt{\log{\left(x \right)} + 1}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                        ____________
         1            \/ 1 + log(x) 
------------------- - --------------
   2   ____________          2      
2*x *\/ 1 + log(x)          x

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

Simplify

The second derivative [src]

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

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

Simplify

4-я производная [src]

                                                                      3             6                   15              36            44    
                                         /        1     \   8 + ------------- + ----------   48 + ------------- + ------------- + ----------
                                       3*|2 + ----------|                   2   1 + log(x)                    3               2   1 + log(x)
        12              ____________     \    1 + log(x)/       (1 + log(x))                      (1 + log(x))    (1 + log(x))              
- -------------- + 24*\/ 1 + log(x)  - ------------------ - ------------------------------ - -----------------------------------------------
    ____________                           ____________                ____________                              ____________               
  \/ 1 + log(x)                          \/ 1 + log(x)             2*\/ 1 + log(x)                          16*\/ 1 + log(x)                
--------------------------------------------------------------------------------------------------------------------------------------------
                                                                      5                                                                     
                                                                     x

$$\frac{- \frac{3 \left(2 + \frac{1}{\log{\left(x \right)} + 1}\right)}{\sqrt{\log{\left(x \right)} + 1}} + 24 \sqrt{\log{\left(x \right)} + 1} - \frac{8 + \frac{6}{\log{\left(x \right)} + 1} + \frac{3}{\left(\log{\left(x \right)} + 1\right)^{2}}}{2 \sqrt{\log{\left(x \right)} + 1}} - \frac{48 + \frac{44}{\log{\left(x \right)} + 1} + \frac{36}{\left(\log{\left(x \right)} + 1\right)^{2}} + \frac{15}{\left(\log{\left(x \right)} + 1\right)^{3}}}{16 \sqrt{\log{\left(x \right)} + 1}} - \frac{12}{\sqrt{\log{\left(x \right)} + 1}}}{x^{5}}$$

Simplify

The third derivative [src]

                                                3             6                          
                                      8 + ------------- + ----------     /        1     \
                                                      2   1 + log(x)   3*|2 + ----------|
      ____________         3              (1 + log(x))                   \    1 + log(x)/
- 6*\/ 1 + log(x)  + -------------- + ------------------------------ + ------------------
                       ____________              ____________               ____________ 
                     \/ 1 + log(x)           8*\/ 1 + log(x)            4*\/ 1 + log(x)  
-----------------------------------------------------------------------------------------
                                             4                                           
                                            x

$$\frac{\frac{3 \left(2 + \frac{1}{\log{\left(x \right)} + 1}\right)}{4 \sqrt{\log{\left(x \right)} + 1}} - 6 \sqrt{\log{\left(x \right)} + 1} + \frac{8 + \frac{6}{\log{\left(x \right)} + 1} + \frac{3}{\left(\log{\left(x \right)} + 1\right)^{2}}}{8 \sqrt{\log{\left(x \right)} + 1}} + \frac{3}{\sqrt{\log{\left(x \right)} + 1}}}{x^{4}}$$

Simplify

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Derivative of sqrt(1+ln(x))/x

The graph:

Piecewise:

The solution