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

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

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
   ________
  /  2     
\/  x  - 1 
-----------
     x
x2−1x\frac{\sqrt{x^{2} - 1}}{x} 
sqrt(x^2 - 1)/x
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x2−1f{\left(x \right)} = \sqrt{x^{2} - 1} and g(x)=xg{\left(x \right)} = x.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Let u=x2−1u = x^{2} - 1.

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ddx(x2−1)\frac{d}{d x} \left(x^{2} - 1\right):

      1. Differentiate x2−1x^{2} - 1 term by term:

        1. The derivative of the constant −1-1 is zero.

        2. Apply the power rule: x2x^{2} goes to 2x2 x

        The result is: 2x2 x

      The result of the chain rule is:

      xx2−1\frac{x}{\sqrt{x^{2} - 1}}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    Now plug in to the quotient rule:

    x2x2−1−x2−1x2\frac{\frac{x^{2}}{\sqrt{x^{2} - 1}} - \sqrt{x^{2} - 1}}{x^{2}}

  2. Now simplify:

    1x2x2−1\frac{1}{x^{2} \sqrt{x^{2} - 1}}

The answer is:

1x2x2−1\frac{1}{x^{2} \sqrt{x^{2} - 1}}

The graph
02468-8-6-4-2-10102.5-2.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                 ________
                /  2     
     1        \/  x  - 1 
----------- - -----------
   ________         2    
  /  2             x     
\/  x  - 1
1x2−1−x2−1x2\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
−x2x2−1−1x2−1−2x2−1+2x2−1x2x\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(x2x2−1−1(x2−1)32+x2x2−1−1x2x2−1+2x2x2−1−2x2−1x4)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
Derivative of sqrt(x^2-1)/x
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