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Derivative of:
Derivative of asin(x)
Derivative of cos(x^3)
Derivative of (cos(x))^3
Derivative of 1/tan(x)
Identical expressions
zero . two *x^ two *exp(x)
0.2 multiply by x squared multiply by exponent of (x)
zero . two multiply by x to the power of two multiply by exponent of (x)
0.2*x2*exp(x)
0.2*x2*expx
0.2*x²*exp(x)
0.2*x to the power of 2*exp(x)
0.2x^2exp(x)
0.2x2exp(x)
0.2x2expx
0.2x^2expx

The derivative of the function/ 0.2*x^2*exp(x)

Derivative of 0.2x^2exp(x)

The solution

You have entered [src]

 2   
x   x
--*e 
5

$$\frac{x^{2}}{5} e^{x}$$

(x^2/5)*exp(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} e^{x}$ and $g{\left(x \right)} = 5$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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: $x^{2} e^{x} + 2 x e^{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $5$ is zero.
Now plug in to the quotient rule:

$\frac{x^{2} e^{x}}{5} + \frac{2 x e^{x}}{5}$
Now simplify:

$\frac{x \left(x + 2\right) e^{x}}{5}$

The answer is:

$\frac{x \left(x + 2\right) e^{x}}{5}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 2  x        x
x *e    2*x*e 
----- + ------
  5       5

$$\frac{x^{2} e^{x}}{5} + \frac{2 x e^{x}}{5}$$

Simplify

The second derivative [src]

/     2      \  x
\2 + x  + 4*x/*e 
-----------------
        5

$$\frac{\left(x^{2} + 4 x + 2\right) e^{x}}{5}$$

Simplify

The third derivative [src]

/     2      \  x
\6 + x  + 6*x/*e 
-----------------
        5

$$\frac{\left(x^{2} + 6 x + 6\right) e^{x}}{5}$$

Simplify