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Derivative of:
Derivative of x^3+2*x^5
Derivative of (3*x-5)^6
Derivative of 1/3*x^3-4*x
Derivative of x²-4
Identical expressions
(x^ two *exp(x))/ two
(x squared multiply by exponent of (x)) divide by 2
(x to the power of two multiply by exponent of (x)) divide by two
(x2*exp(x))/2
x2*expx/2
(x²*exp(x))/2
(x to the power of 2*exp(x))/2
(x^2exp(x))/2
(x2exp(x))/2
x2expx/2
x^2expx/2
(x^2*exp(x)) divide by 2

The derivative of the function/ (x^2*exp(x))/2

Derivative of (x^2*exp(x))/2

The solution

You have entered [src]

 2  x
x *e 
-----
  2

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

(x^2*exp(x))/2

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

3-я производная [src]

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

$$\frac{\left(x^{2} + 6 x + 6\right) e^{x}}{2}$$

Simplify

The third derivative [src]

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

$$\frac{\left(x^{2} + 6 x + 6\right) e^{x}}{2}$$

Simplify