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Derivative of (3*log(x))/sqrt(x)

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3*log(x)
--------
   ___  
 \/ x

$$\frac{3 \log{\left(x \right)}}{\sqrt{x}}$$

d /3*log(x)\
--|--------|
dx|   ___  |
  \ \/ x   /

$$\frac{d}{d x} \frac{3 \log{\left(x \right)}}{\sqrt{x}}$$

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3      3*log(x)
------- - --------
    ___       3/2 
x*\/ x     2*x

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

Simplify

The second derivative [src]

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

$$\frac{3 \cdot \left(\frac{3 \log{\left(x \right)}}{4} - 2\right)}{x^{\frac{5}{2}}}$$

Simplify

The third derivative [src]

3*(46 - 15*log(x))
------------------
         7/2      
      8*x

$$\frac{3 \cdot \left(46 - 15 \log{\left(x \right)}\right)}{8 x^{\frac{7}{2}}}$$

Simplify

The graph