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Derivative of:
Derivative of sin(2*x^2+3)
Derivative of 5*x^4-7*x^2-x
Derivative of 4*sqrt(3)*x/3
Derivative of 3/x^5
Graphing y =:
x^3/e^x
Integral of d{x}:
x^3/e^x
Identical expressions
x^ three /e^x
x cubed divide by e to the power of x
x to the power of three divide by e to the power of x
x3/ex
x³/e^x
x to the power of 3/e to the power of x
x^3 divide by e^x

The derivative of the function/ x^3/e^x

Derivative of x^3/e^x

The solution

You have entered [src]

 3
x 
--
 x
E

$$\frac{x^{3}}{e^{x}}$$

x^3/E^x

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{3}$ and $g{\left(x \right)} = e^{x}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
Now plug in to the quotient rule:

$\left(- x^{3} e^{x} + 3 x^{2} e^{x}\right) e^{- 2 x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   3  -x      2  -x
- x *e   + 3*x *e

$$- x^{3} e^{- x} + 3 x^{2} e^{- x}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/     3             2\  -x
\6 - x  - 18*x + 9*x /*e

$$\left(- x^{3} + 9 x^{2} - 18 x + 6\right) e^{- x}$$

Simplify

The graph

Derivative of x^3/e^x