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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
Identical expressions
ax^ two e^(-x/2)
ax squared e to the power of ( minus x divide by 2)
ax to the power of two e to the power of ( minus x divide by 2)
ax2e(-x/2)
ax2e-x/2
ax²e^(-x/2)
ax to the power of 2e to the power of (-x/2)
ax^2e^-x/2
ax^2e^(-x divide by 2)
Similar expressions
ax^2e^(x/2)
What you mean?
a*x^2*e^(-x/2)
a*x^((2*e)^(-x/2))
a*x^2*e^(-x/2)
a*x^((2*e)^(-x/2))
a*x^((2*e)^(-x/2))

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

You entered:

ax^2e^(-x/2)

What you mean?

a*x^2*e^(-x/2)

a*x^((2*e)^(-x/2))

a*x^2*e^(-x/2)

a*x^((2*e)^(-x/2))

a*x^((2*e)^(-x/2))

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

The solution

You have entered [src]

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

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

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

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

Transform

Detail solution

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

$\frac{a x \left(4 - x\right) e^{- \frac{x}{2}}}{2}$

The answer is:

$\frac{a x \left(4 - x\right) e^{- \frac{x}{2}}}{2}$

The first derivative [src]

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

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

Simplify

The second derivative [src]

                  -x 
  /           2\  ---
  |          x |   2 
a*|2 - 2*x + --|*e   
  \          4 /

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

Simplify

The third derivative [src]

                   -x 
  /      2      \  ---
  |     x    3*x|   2 
a*|-3 - -- + ---|*e   
  \     8     2 /

$$a \left(- \frac{x^{2}}{8} + \frac{3 x}{2} - 3\right) e^{- \frac{x}{2}}$$

Simplify