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Derivative of:
Derivative of y=2x+1
Derivative of asin(t)
Derivative of 7/x^4
Derivative of x*sin(4*x)
Integral of d{x}:
x^2/e^(x^2)
Identical expressions
x^ two /e^(x^ two)
x squared divide by e to the power of (x squared )
x to the power of two divide by e to the power of (x to the power of two)
x2/e(x2)
x2/ex2
x²/e^(x²)
x to the power of 2/e to the power of (x to the power of 2)
x^2/e^x^2
x^2 divide by e^(x^2)

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

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

The solution

You have entered [src]

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

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

x^2/E^(x^2)

Detail solution

Apply the quotient rule, which is:

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

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

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

$\left(- 2 x^{3} e^{x^{2}} + 2 x e^{x^{2}}\right) e^{- 2 x^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

$$- 2 x^{3} e^{- x^{2}} + 2 x e^{- x^{2}}$$

Simplify

The second derivative [src]

                                 2
  /       2    2 /        2\\  -x 
2*\1 - 4*x  + x *\-1 + 2*x //*e

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

Simplify

The third derivative [src]

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

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

Simplify