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Similar expressions
x/(√(1-x^2))

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

Derivative of x/(√(1+x^2))

The solution

You have entered [src]

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

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

d /     x     \
--|-----------|
dx|   ________|
  |  /      2 |
  \\/  1 + x  /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                    2    
     1             x     
----------- - -----------
   ________           3/2
  /      2    /     2\   
\/  1 + x     \1 + x /

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

Simplify

The second derivative [src]

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

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

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 /

$$\frac{3 \left(- \frac{x^{2} \cdot \left(\frac{5 x^{2}}{x^{2} + 1} - 3\right)}{x^{2} + 1} + \frac{3 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{\frac{3}{2}}}$$

Simplify

The graph

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Derivative of x/(√(1+x^2))

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The solution