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Derivative of:
Derivative of 1/3*x^3
Derivative of x*x^(1/2)
Derivative of x^(5/2)
Derivative of (x-4)^2
Integral of d{x}:
e^((-x^2)/2)
Identical expressions
e^((-x^ two)/ two)
e to the power of (( minus x squared ) divide by 2)
e to the power of (( minus x to the power of two) divide by two)
e((-x2)/2)
e-x2/2
e^((-x²)/2)
e to the power of ((-x to the power of 2)/2)
e^-x^2/2
e^((-x^2) divide by 2)
Similar expressions
e^((x^2)/2)

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

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

The solution

You have entered [src]

   2 
 -x  
 ----
  2  
E

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

E^((-x^2)/2)

Detail solution

Let $u = \frac{\left(-1\right) x^{2}}{2}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\left(-1\right) x^{2}}{2}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  So, the result is: $- x$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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