Derivative of lnx/2√x+√x/x

The solution

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

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

(log(x)/2)*sqrt(x) + sqrt(x)/x

Detail solution

Differentiate $\sqrt{x} \frac{\log{\left(x \right)}}{2} + \frac{\sqrt{x}}{x}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sqrt{x} \log{\left(x \right)}$ and $g{\left(x \right)} = 2$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
    $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    The result is: $\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of the constant $2$ is zero.
  Now plug in to the quotient rule:
  
  $\frac{\log{\left(x \right)}}{4 \sqrt{x}} + \frac{1}{2 \sqrt{x}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sqrt{x}$ and $g{\left(x \right)} = x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  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:
  
  $- \frac{1}{2 x^{\frac{3}{2}}}$
The result is: $\frac{\log{\left(x \right)}}{4 \sqrt{x}} + \frac{1}{2 \sqrt{x}} - \frac{1}{2 x^{\frac{3}{2}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

          6
-log(x) + -
          x
-----------
      3/2  
   8*x

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

Simplify

The third derivative [src]

     30           
-2 - -- + 3*log(x)
     x            
------------------
         5/2      
     16*x

$$\frac{3 \log{\left(x \right)} - 2 - \frac{30}{x}}{16 x^{\frac{5}{2}}}$$

Simplify

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Derivative of lnx/2√x+√x/x

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The solution