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(x)*(exp(-x^2))

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Identical expressions
(x)*(exp(-x^ two))
(x) multiply by ( exponent of ( minus x squared ))
(x) multiply by ( exponent of ( minus x to the power of two))
(x)*(exp(-x2))
x*exp-x2
(x)*(exp(-x²))
(x)*(exp(-x to the power of 2))
(x)(exp(-x^2))
(x)(exp(-x2))
xexp-x2
xexp-x^2
Similar expressions
x*exp(-x^2)
(x)*(exp(x^2))

The derivative of the function/ (x)*(exp(-x^2))

Derivative of (x)*(exp(-x^2))

The solution

You have entered [src]

     2
   -x 
x*e

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
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:
  
  $2 x e^{x^{2}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

          2      2
     2  -x     -x 
- 2*x *e    + e

$$- 2 x^{2} e^{- x^{2}} + e^{- x^{2}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                    2
  /        2      2 /        2\\  -x 
2*\-3 + 6*x  - 2*x *\-3 + 2*x //*e

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

Simplify

The graph

Derivative of (x)*(exp(-x^2))