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

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

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

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from to

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The solution

You have entered [src] 
     x     
-----------
   ________
  /  2     
\/  x  - 1
xx2−1\frac{x}{\sqrt{x^{2} - 1}} 
d /     x     \
--|-----------|
dx|   ________|
  |  /  2     |
  \\/  x  - 1 /
ddxxx2−1\frac{d}{d x} \frac{x}{\sqrt{x^{2} - 1}}
Transform
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)=xf{\left(x \right)} = x and g(x)=x2−1g{\left(x \right)} = \sqrt{x^{2} - 1}.

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

    1. Apply the power rule: xx goes to 11

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

    Now plug in to the quotient rule:

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

  2. Now simplify:

    −1(x2−1)32- \frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}

The answer is:

−1(x2−1)32- \frac{1}{\left(x^{2} - 1\right)^{\frac{3}{2}}}

The graph
02468-8-6-4-2-1010-1010
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                    2    
     1             x     
----------- - -----------
   ________           3/2
  /  2        / 2    \   
\/  x  - 1    \x  - 1/
−x2(x2−1)32+1x2−1- \frac{x^{2}}{\left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{2} - 1}}
Simplify
The second derivative [src] 
  /          2 \
  |       3*x  |
x*|-3 + -------|
  |           2|
  \     -1 + x /
----------------
           3/2  
  /      2\     
  \-1 + x /
x(3x2x2−1−3)(x2−1)32\frac{x \left(\frac{3 x^{2}}{x^{2} - 1} - 3\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}
Simplify
The third derivative [src] 
  /                  /          2 \\
  |                2 |       5*x  ||
  |               x *|-3 + -------||
  |          2       |           2||
  |       3*x        \     -1 + x /|
3*|-1 + ------- - -----------------|
  |           2              2     |
  \     -1 + x         -1 + x      /
------------------------------------
                     3/2            
            /      2\               
            \-1 + x /
3(−x2⋅(5x2x2−1−3)x2−1+3x2x2−1−1)(x2−1)32\frac{3 \left(- \frac{x^{2} \cdot \left(\frac{5 x^{2}}{x^{2} - 1} - 3\right)}{x^{2} - 1} + \frac{3 x^{2}}{x^{2} - 1} - 1\right)}{\left(x^{2} - 1\right)^{\frac{3}{2}}}
Simplify
The graph
Derivative of x/sqrt(x^2-1)
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