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Derivative of:
Derivative of x^4*sin(x)
Derivative of 3*x
Derivative of sin(y)
Derivative of 1/x²
Identical expressions
- two xe^(-x^2)
minus 2xe to the power of ( minus x squared )
minus two xe to the power of ( minus x squared )
-2xe(-x2)
-2xe-x2
-2xe^(-x²)
-2xe to the power of (-x to the power of 2)
-2xe^-x^2
Similar expressions
-2xe^(x^2)
2xe^(-x^2)

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

Derivative of -2xe^(-x^2)

The solution

You have entered [src]

        2
      -x 
-2*x*e

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

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

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

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. 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:
  
  $\left(- 2 x^{2} e^{x^{2}} + e^{x^{2}}\right) e^{- 2 x^{2}}$
So, the result is: $- 2 \left(- 2 x^{2} e^{x^{2}} + e^{x^{2}}\right) e^{- 2 x^{2}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

       2           2
     -x       2  -x 
- 2*e    + 4*x *e

$$4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of -2xe^(-x^2)