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The derivative of the function/ log(1/(x*x-2)^(1/2))

Derivative of log(1/(x*x-2)^(1/2))

The solution

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   /     1     \
log|-----------|
   |  _________|
   \\/ x*x - 2 /

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

log(1/(sqrt(x*x - 2)))

Detail solution

Let $u = \frac{1}{\sqrt{x x - 2}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{\sqrt{x x - 2}}$ :
1. Let $u = \sqrt{x x - 2}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x x - 2}$ :
  1. Let $u = x x - 2$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x x - 2\right)$ :
    1. Differentiate $x x - 2$ term by term:
      1. Apply the product rule:
        
        $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
        
        $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
        
        Apply the power rule: $x$ goes to $1$
        
        $g{\left(x \right)} = x$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
        
        Apply the power rule: $x$ goes to $1$
        
        The result is: $2 x$
      2. The derivative of the constant $-2$ is zero.
      The result is: $2 x$
    The result of the chain rule is:
    
    $\frac{x}{\sqrt{x x - 2}}$
  The result of the chain rule is:
  
  $- \frac{x}{\left(x x - 2\right)^{\frac{3}{2}}}$
The result of the chain rule is:

$- \frac{x}{x x - 2}$
Now simplify:

$- \frac{x}{x^{2} - 2}$

The answer is:

$- \frac{x}{x^{2} - 2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  -x   
-------
x*x - 2

$$- \frac{x}{x x - 2}$$

Simplify

The second derivative [src]

          2 
       2*x  
-1 + -------
           2
     -2 + x 
------------
        2   
  -2 + x

$$\frac{\frac{2 x^{2}}{x^{2} - 2} - 1}{x^{2} - 2}$$

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of log(1/(x*x-2)^(1/2))

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