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Derivative of:
Derivative of 4/x^3
Derivative of x^6/6
Derivative of x^3/x
Derivative of (x-3)^4
Integral of d{x}:
1/sqrt(x^2+1)
Identical expressions
one /sqrt(x^ two + one)
1 divide by square root of (x squared plus 1)
one divide by square root of (x to the power of two plus one)
1/√(x^2+1)
1/sqrt(x2+1)
1/sqrtx2+1
1/sqrt(x²+1)
1/sqrt(x to the power of 2+1)
1/sqrtx^2+1
1 divide by sqrt(x^2+1)
Similar expressions
1/sqrt(x^2-1)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Let $u = \sqrt{x^{2} + 1}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x^{2} + 1}$ :
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. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $1$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{x}{\sqrt{x^{2} + 1}}$
The result of the chain rule is:

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

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

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     
\x  + 1/*\/  x  + 1

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

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

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