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one /(sqrt(x)-ln(x+ one))
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Similar expressions
1/(sqrt(x)+ln(x+1))
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The derivative of the function/ 1/(sqrt(x)-ln(x+1))

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

The solution

You have entered [src]

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

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

1/(sqrt(x) - log(x + 1))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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

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The solution