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e^(x^2)/x
Graphing y =:
e^(x^2)/x
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The derivative of the function/ e^(x^2)/x

Derivative of e^(x^2)/x

The solution

You have entered [src]

 / 2\
 \x /
E    
-----
  x

$$\frac{e^{x^{2}}}{x}$$

E^(x^2)/x

Detail solution

Apply the quotient rule, which is:

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

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

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

$\frac{2 x^{2} e^{x^{2}} - e^{x^{2}}}{x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           / 2\
   / 2\    \x /
   \x /   e    
2*e     - -----
             2 
            x

$$2 e^{x^{2}} - \frac{e^{x^{2}}}{x^{2}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                       /       2\\  / 2\
  |    3       2   6    3*\1 + 2*x /|  \x /
2*|6 - -- + 4*x  + -- - ------------|*e    
  |     4           2         2     |      
  \    x           x         x      /

$$2 \left(4 x^{2} + 6 - \frac{3 \left(2 x^{2} + 1\right)}{x^{2}} + \frac{6}{x^{2}} - \frac{3}{x^{4}}\right) e^{x^{2}}$$

Simplify

The graph

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Identical expressions

Derivative of e^(x^2)/x

The graph:

Piecewise:

The solution