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

You entered:

sqrt(x^2+1)/x

What you mean?

sqrt(x^2 + 1)/x
 
sqrt(x*(2 + 1))/x
 
sqrt(x^(2 + 1))/x
 
sqrt(x^2 + 1)/x
 
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} 
  /   ________\
  |  /  2     |
d |\/  x  + 1 |
--|-----------|
dx\     x     /
ddxx2+1x\frac{d}{d x} \frac{\sqrt{x^{2} + 1}}{x}
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)=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 11 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
02468-8-6-4-2-1010-100100
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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