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Limit of the function:
1/sqrt(1+x^2)
Integral of d{x}:
1/sqrt(1+x^2)
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Similar expressions
1/sqrt(1-x^2)

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

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

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

     /         2 \
     |      5*x  |
-3*x*|-3 + ------|
     |          2|
     \     1 + x /
------------------
           5/2    
   /     2\       
   \1 + x /

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

Simplify

The graph

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

The graph:

Piecewise:

The solution