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Derivative of ln(x)/(sqrt(x+2*x))

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

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

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

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = \sqrt{3} \sqrt{x}$ .

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. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  So, the result is: $\frac{\sqrt{3}}{2 \sqrt{x}}$
Now plug in to the quotient rule:

$\frac{- \frac{\sqrt{3} \log{\left(x \right)}}{2 \sqrt{x}} + \frac{\sqrt{3}}{\sqrt{x}}}{3 x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  ___                
\/ 3 *(-8 + 3*log(x))
---------------------
           5/2       
       12*x

$$\frac{\sqrt{3} \left(3 \log{\left(x \right)} - 8\right)}{12 x^{\frac{5}{2}}}$$

Simplify

The third derivative [src]

  ___                 
\/ 3 *(46 - 15*log(x))
----------------------
           7/2        
       24*x

$$\frac{\sqrt{3} \left(46 - 15 \log{\left(x \right)}\right)}{24 x^{\frac{7}{2}}}$$

Simplify