Mister Exam

Other calculators

You entered:

ax^2e^(-x/2)

What you mean?

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

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

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           
$$\frac{\partial}{\partial x} a x^{2} e^{\frac{\left(-1\right) x}{2}}$$
Detail solution
  1. The derivative of a constant times a function is the constant times the derivative of the function.

    1. Apply the product rule:

      ; to find :

      1. Let .

      2. The derivative of is itself.

      3. Then, apply the chain rule. Multiply by :

        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: goes to

            So, the result is:

          So, the result is:

        The result of the chain rule is:

      ; to find :

      1. Apply the power rule: goes to

      The result is:

    So, the result is:

  2. Now simplify:


The answer is:

The first derivative [src]
                   -x 
       -x          ---
       ---      2   2 
        2    a*x *e   
2*a*x*e    - ---------
                 2    
$$- \frac{a x^{2} e^{\frac{\left(-1\right) x}{2}}}{2} + 2 a x e^{\frac{\left(-1\right) x}{2}}$$
The second derivative [src]
                  -x 
  /           2\  ---
  |          x |   2 
a*|2 - 2*x + --|*e   
  \          4 /     
$$a \left(\frac{x^{2}}{4} - 2 x + 2\right) e^{- \frac{x}{2}}$$
The third derivative [src]
                   -x 
  /      2      \  ---
  |     x    3*x|   2 
a*|-3 - -- + ---|*e   
  \     8     2 /     
$$a \left(- \frac{x^{2}}{8} + \frac{3 x}{2} - 3\right) e^{- \frac{x}{2}}$$