Derivative of ln(x^(1÷2)+2)

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   /  ___    \
log\\/ x  + 2/

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

log(sqrt(x) + 2)

Detail solution

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

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

         1         
-------------------
    ___ /  ___    \
2*\/ x *\\/ x  + 2/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 3             2                 3       
---- + ----------------- + --------------
 5/2                   2    2 /      ___\
x       3/2 /      ___\    x *\2 + \/ x /
       x   *\2 + \/ x /                  
-----------------------------------------
                /      ___\              
              8*\2 + \/ x /

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

Simplify

The graph

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Derivative of ln(x^(1÷2)+2)

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The solution