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

Derivative of (e^(-x^2)-1)/x

The solution

You have entered [src]

   2    
 -x     
e    - 1
--------
   x

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

  /   2    \
  | -x     |
d |e    - 1|
--|--------|
dx\   x    /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $1 - e^{x^{2}}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    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:
      
      $2 x e^{x^{2}}$
    So, the result is: $- 2 x e^{x^{2}}$
  The result is: $- 2 x e^{x^{2}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = e^{x^{2}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $2 x e^{x^{2}}$
  The result is: $2 x^{2} e^{x^{2}} + e^{x^{2}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              2    
       2    -x     
     -x    e    - 1
- 2*e    - --------
               2   
              x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (e^(-x^2)-1)/x

The graph:

Piecewise:

The solution