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Identical expressions
y=2ln((x+ three)/x)- three
y equally 2ln((x plus 3) divide by x) minus 3
y equally 2ln((x plus three) divide by x) minus three
y=2lnx+3/x-3
y=2ln((x+3) divide by x)-3
Similar expressions
y=2ln((x-3)/x)-3
y=2ln((x+3)/x)+3

The derivative of the function/ y=2ln((x+3)/x)-3

Derivative of y=2ln((x+3)/x)-3

The solution

You have entered [src]

     /x + 3\    
2*log|-----| - 3
     \  x  /

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

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

Detail solution

Differentiate $2 \log{\left(\frac{x + 3}{x} \right)} - 3$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{x + 3}{x}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x + 3}{x}$ :
    1. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = x + 3$ and $g{\left(x \right)} = x$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Differentiate $x + 3$ term by term:
        
        The derivative of the constant $3$ is zero.
        
        Apply the power rule: $x$ goes to $1$
        
        The result is: $1$
      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:
      
      $- \frac{3}{x^{2}}$
    The result of the chain rule is:
    
    $- \frac{3}{x \left(x + 3\right)}$
  So, the result is: $- \frac{6}{x \left(x + 3\right)}$
2. The derivative of the constant $-3$ is zero.
The result is: $- \frac{6}{x \left(x + 3\right)}$
Now simplify:

$- \frac{6}{x \left(x + 3\right)}$

The answer is:

$- \frac{6}{x \left(x + 3\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of y=2ln((x+3)/x)-3

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The solution