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

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

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

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The solution

You have entered [src] 
       1     
1*-----------
     ________
    /      2 
  \/  1 + x
1⋅1x2+11 \cdot \frac{1}{\sqrt{x^{2} + 1}} 
d /       1     \
--|1*-----------|
dx|     ________|
  |    /      2 |
  \  \/  1 + x  /
ddx1⋅1x2+1\frac{d}{d x} 1 \cdot \frac{1}{\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)=1f{\left(x \right)} = 1 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. The derivative of the constant 11 is zero.

    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 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}}

    Now plug in to the quotient rule:

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

The answer is:

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

The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        -x          
--------------------
            ________
/     2\   /      2 
\1 + x /*\/  1 + x
−xx2+1(x2+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 /
3x2x2+1−1(x2+1)32\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 /
−3x(5x2x2+1−3)(x2+1)52- \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(1+x^2)
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