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Derivative of:
Derivative of 2*x-1
Derivative of i*n*sin(x)
Derivative of cosx/1+sinx
Derivative of 1/(2x)
Identical expressions
y=x*sqrt((x^ two - one)/(x^ two + one))
y equally x multiply by square root of ((x squared minus 1) divide by (x squared plus 1))
y equally x multiply by square root of ((x to the power of two minus one) divide by (x to the power of two plus one))
y=x*√((x^2-1)/(x^2+1))
y=x*sqrt((x2-1)/(x2+1))
y=x*sqrtx2-1/x2+1
y=x*sqrt((x²-1)/(x²+1))
y=x*sqrt((x to the power of 2-1)/(x to the power of 2+1))
y=xsqrt((x^2-1)/(x^2+1))
y=xsqrt((x2-1)/(x2+1))
y=xsqrtx2-1/x2+1
y=xsqrtx^2-1/x^2+1
y=x*sqrt((x^2-1) divide by (x^2+1))
Similar expressions
y=x*sqrt((x^2-1)/(x^2-1))
y=x*sqrt((x^2+1)/(x^2+1))

The derivative of the function/ y=x*sqrt((x^2-1)/(x^2+1))

Derivative of y=x*sqrt((x^2-1)/(x^2+1))

The solution

You have entered [src]

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

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

x*sqrt((x^2 - 1)/(x^2 + 1))

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
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 $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sqrt{x^{2} - 1}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $\frac{x}{\sqrt{x^{2} - 1}}$
  The result is: $\frac{x^{2}}{\sqrt{x^{2} - 1}} + \sqrt{x^{2} - 1}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $\frac{x}{\sqrt{x^{2} + 1}}$
Now plug in to the quotient rule:

$\frac{- \frac{x^{2} \sqrt{x^{2} - 1}}{\sqrt{x^{2} + 1}} + \sqrt{x^{2} + 1} \left(\frac{x^{2}}{\sqrt{x^{2} - 1}} + \sqrt{x^{2} - 1}\right)}{x^{2} + 1}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /                                                                                /                                   _________           _________                        \   /                    _________\                                                                                                                                                                                                      /                    _________\                                                                        /                    _________\\                                                                                                                 /                    _________\
   |    /          2       2       2 /      2\\                      /           2\ |                       2          /       2       2   /       2                2        |   |                   /       2 | /          2       2       2 /      2\\      /          2       2       2 /      2\\                                                   /          2       2       2 /      2\\        /           2\ |                   /       2 |                                                         /           2\ |                   /       2 ||     /          2       2       2 /      2\\                                                      /           2\ |                   /       2 |
   |    |    -1 + x     4*x     4*x *\-1 + x /|     /           2\   |     -1 + x | |       1              x         \/  -1 + x     3*x *\/  -1 + x              2*x         |   |       1         \/  -1 + x  | |    -1 + x     4*x     4*x *\-1 + x /|      |    -1 + x     2*x     2*x *\-1 + x /|        /           2\        /           2\       |    -1 + x     4*x     4*x *\-1 + x /|      2 |     -1 + x | |       1         \/  -1 + x  |         /           2\           /           2\       2 |     -1 + x | |       1         \/  -1 + x  ||     |    -1 + x     4*x     4*x *\-1 + x /|        /           2\         /           2\       2 |     -1 + x | |       1         \/  -1 + x  |
   |  3*|1 - ------- - ------ + --------------|     |     -1 + x |   |-1 + -------|*|- ------------ + ------------ + ------------ - ----------------- + ---------------------|   |- ------------ + ------------|*|1 - ------- - ------ + --------------|   12*|1 - ------- - ------ + --------------|      2 |     -1 + x |        |     -1 + x |     3*|1 - ------- - ------ + --------------|   2*x *|-1 + -------|*|- ------------ + ------------|       2 |     -1 + x |         2 |     -1 + x |    2*x *|-1 + -------|*|- ------------ + ------------||   3*|1 - ------- - ------ + --------------|      2 |     -1 + x |       2 |     -1 + x |    3*x *|-1 + -------|*|- ------------ + ------------|
   |    |          2        2             2   |   2*|-1 + -------|   |           2| |     _________            3/2           2                  2                   _________|   |     _________           2   | |          2        2             2   |      |          2        2             2   |   6*x *|-1 + -------|      2*|-1 + -------|       |          2        2             2   |        |           2| |     _________           2   |    2*x *|-1 + -------|      4*x *|-1 + -------|         |           2| |     _________           2   ||     |          2        2             2   |   6*x *|-1 + -------|    6*x *|-1 + -------|         |           2| |     _________           2   |
   |    |     1 + x    1 + x      /     2\    |     |           2|   \      1 + x / |    /       2    /      2\         1 + x           /     2\        /     2\   /       2 |   |    /       2       1 + x    | |     1 + x    1 + x      /     2\    |      |     1 + x    1 + x      /     2\    |        |           2|        |           2|       |     1 + x    1 + x      /     2\    |        \      1 + x / |    /       2       1 + x    |         |           2|           |           2|         \      1 + x / |    /       2       1 + x    ||     |     1 + x    1 + x      /     2\    |        |           2|         |           2|         \      1 + x / |    /       2       1 + x    |
 2 |    \                         \1 + x /    /     \      1 + x /                  \  \/  -1 + x     \-1 + x /                         \1 + x /        \1 + x /*\/  -1 + x  /   \  \/  -1 + x                 / \                         \1 + x /    /      \                         \1 + x /    /        \      1 + x /        \      1 + x /       \                         \1 + x /    /                       \  \/  -1 + x                 /         \      1 + x /           \      1 + x /                        \  \/  -1 + x                 /|     \                         \1 + x /    /        \      1 + x /         \      1 + x /                        \  \/  -1 + x                 /
x *|- ----------------------------------------- + ---------------- + --------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------- - ------------------------------------------ - ------------------- - --------------------- + ----------------------------------------- - --------------------------------------------------- + ---------------------- + --------------------- + ---------------------------------------------------| + ----------------------------------------- + ------------------- - --------------------- + ---------------------------------------------------
   |                          3/2                            3/2                                                            2                                                                                          2                                                         _________                           5/2                  _________                         _________                                           2                               2    _________                     3/2                    /     2\ /      2\                |                     _________                              3/2                  _________                               2                      
   |                 /      2\                      /      2\                                                         -1 + x                                                                                     -1 + x                                              /     2\   /       2                   /      2\         /     2\   /       2              /     2\   /       2                                   /      2\                        /     2\    /       2    /     2\ /      2\                       \1 + x /*\-1 + x /                |                    /       2                      /      2\         /     2\   /       2                          -1 + x                       
   \                 \-1 + x /                      \-1 + x /                                                                                                                                                                                                        \1 + x /*\/  -1 + x                    \-1 + x /         \1 + x /*\/  -1 + x               \1 + x /*\/  -1 + x                                    \-1 + x /                        \1 + x / *\/  -1 + x     \1 + x /*\-1 + x /                                                         /                  \/  -1 + x                       \-1 + x /         \1 + x /*\/  -1 + x                                                        
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                                                                                                                                                                                                                                                                                                                                                    ________                                                                                                                                                                                                                                                                                                                                                 
                                                                                                                                                                                                                                                                                                                                                   /      2                                                                                                                                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                                                                                                                                 \/  1 + x

$$\frac{x^{2} \left(\frac{2 x^{2} \left(\frac{\sqrt{x^{2} - 1}}{x^{2} + 1} - \frac{1}{\sqrt{x^{2} - 1}}\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right) \left(x^{2} + 1\right)} - \frac{2 x^{2} \left(\frac{\sqrt{x^{2} - 1}}{x^{2} + 1} - \frac{1}{\sqrt{x^{2} - 1}}\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right)^{2}} + \frac{2 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\sqrt{x^{2} - 1} \left(x^{2} + 1\right)^{2}} + \frac{4 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}} \left(x^{2} + 1\right)} - \frac{6 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{5}{2}}} - \frac{\left(\frac{\sqrt{x^{2} - 1}}{x^{2} + 1} - \frac{1}{\sqrt{x^{2} - 1}}\right) \left(\frac{4 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{4 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{x^{2} - 1} + \frac{\left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right) \left(- \frac{3 x^{2} \sqrt{x^{2} - 1}}{\left(x^{2} + 1\right)^{2}} + \frac{2 x^{2}}{\sqrt{x^{2} - 1} \left(x^{2} + 1\right)} + \frac{x^{2}}{\left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{\sqrt{x^{2} - 1}}{x^{2} + 1} - \frac{1}{\sqrt{x^{2} - 1}}\right)}{x^{2} - 1} - \frac{2 \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\sqrt{x^{2} - 1} \left(x^{2} + 1\right)} - \frac{12 \left(\frac{2 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{2 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{\sqrt{x^{2} - 1} \left(x^{2} + 1\right)} + \frac{3 \left(\frac{4 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{4 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{\sqrt{x^{2} - 1} \left(x^{2} + 1\right)} + \frac{2 \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{3 \left(\frac{4 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{4 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}\right) + \frac{3 x^{2} \left(\frac{\sqrt{x^{2} - 1}}{x^{2} + 1} - \frac{1}{\sqrt{x^{2} - 1}}\right) \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{x^{2} - 1} - \frac{6 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\sqrt{x^{2} - 1} \left(x^{2} + 1\right)} + \frac{6 x^{2} \left(\frac{x^{2} - 1}{x^{2} + 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{3 \left(\frac{4 x^{2} \left(x^{2} - 1\right)}{\left(x^{2} + 1\right)^{2}} - \frac{4 x^{2}}{x^{2} + 1} - \frac{x^{2} - 1}{x^{2} + 1} + 1\right)}{\sqrt{x^{2} - 1}}}{\sqrt{x^{2} + 1}}$$

Simplify

The graph

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How to use it?

Identical expressions

Similar expressions

Derivative of y=x*sqrt((x^2-1)/(x^2+1))

The graph:

Piecewise:

The solution