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Integral of d{x}:
sqrt(x^2-1)/x
Identical expressions
sqrt(x^ two - one)/x
square root of (x squared minus 1) divide by x
square root of (x to the power of two minus one) divide by x
√(x^2-1)/x
sqrt(x2-1)/x
sqrtx2-1/x
sqrt(x²-1)/x
sqrt(x to the power of 2-1)/x
sqrtx^2-1/x
sqrt(x^2-1) divide by x
Similar expressions
sqrt(x^2+1)/x

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

Derivative of sqrt(x^2-1)/x

The solution

You have entered [src]

   ________
  /  2     
\/  x  - 1 
-----------
     x

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

sqrt(x^2 - 1)/x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sqrt{x^{2} - 1}$ and $g{\left(x \right)} = x$ .

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

$\frac{\frac{x^{2}}{\sqrt{x^{2} - 1}} - \sqrt{x^{2} - 1}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 ________
                /  2     
     1        \/  x  - 1 
----------- - -----------
   ________         2    
  /  2             x     
\/  x  - 1

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

Simplify

The second derivative [src]

                          2                   
                         x                    
                 -1 + -------        _________
                            2       /       2 
       2              -1 + x    2*\/  -1 + x  
- ------------ - ------------ + --------------
     _________      _________          2      
    /       2      /       2          x       
  \/  -1 + x     \/  -1 + x                   
----------------------------------------------
                      x

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

Simplify

The third derivative [src]

  /         2                                                   2   \
  |        x                                                   x    |
  |-1 + -------        _________                       -1 + ------- |
  |           2       /       2                                   2 |
  |     -1 + x    2*\/  -1 + x            2                 -1 + x  |
3*|------------ - -------------- + --------------- + ---------------|
  |         3/2          4               _________         _________|
  |/      2\            x           2   /       2     2   /       2 |
  \\-1 + x /                       x *\/  -1 + x     x *\/  -1 + x  /

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

Simplify

The graph

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Derivative of sqrt(x^2-1)/x

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The solution