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Identical expressions
е^(- two x)/(one +x^ two)^(one /2)
е to the power of ( minus 2x) divide by (1 plus x squared ) to the power of (1 divide by 2)
е to the power of ( minus two x) divide by (one plus x to the power of two) to the power of (one divide by 2)
е(-2x)/(1+x2)(1/2)
е-2x/1+x21/2
е^(-2x)/(1+x²)^(1/2)
е to the power of (-2x)/(1+x to the power of 2) to the power of (1/2)
е^-2x/1+x^2^1/2
е^(-2x) divide by (1+x^2)^(1 divide by 2)
Similar expressions
е^(-2x)/(1-x^2)^(1/2)
е^(2x)/(1+x^2)^(1/2)

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

Derivative of е^(-2x)/(1+x^2)^(1/2)

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sqrt{x^{2} + 1}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  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:
    
    $\frac{x}{\sqrt{x^{2} + 1}}$
  $g{\left(x \right)} = e^{2 x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 x$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $2 e^{2 x}$
  The result is: $\frac{x e^{2 x}}{\sqrt{x^{2} + 1}} + 2 \sqrt{x^{2} + 1} e^{2 x}$
Now plug in to the quotient rule:

$\frac{\left(- \frac{x e^{2 x}}{\sqrt{x^{2} + 1}} - 2 \sqrt{x^{2} + 1} e^{2 x}\right) e^{- 4 x}}{x^{2} + 1}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/             2          \      
|          3*x           |      
|    -1 + ------         |      
|              2         |      
|         1 + x     4*x  |  -2*x
|4 + ----------- + ------|*e    
|            2          2|      
\       1 + x      1 + x /      
--------------------------------
             ________           
            /      2            
          \/  1 + x

$$\frac{\left(\frac{4 x}{x^{2} + 1} + 4 + \frac{\frac{3 x^{2}}{x^{2} + 1} - 1}{x^{2} + 1}\right) e^{- 2 x}}{\sqrt{x^{2} + 1}}$$

Simplify

The third derivative [src]

 /      /         2 \                /         2 \\       
 |      |      3*x  |                |      5*x  ||       
 |    6*|-1 + ------|            3*x*|-3 + ------||       
 |      |          2|                |          2||       
 |      \     1 + x /    12*x        \     1 + x /|  -2*x 
-|8 + --------------- + ------ + -----------------|*e     
 |              2            2               2    |       
 |         1 + x        1 + x        /     2\     |       
 \                                   \1 + x /     /       
----------------------------------------------------------
                          ________                        
                         /      2                         
                       \/  1 + x

$$- \frac{\left(\frac{12 x}{x^{2} + 1} + \frac{3 x \left(\frac{5 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{2}} + 8 + \frac{6 \cdot \left(\frac{3 x^{2}}{x^{2} + 1} - 1\right)}{x^{2} + 1}\right) e^{- 2 x}}{\sqrt{x^{2} + 1}}$$

Simplify

The graph

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Derivative of е^(-2x)/(1+x^2)^(1/2)

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The solution