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Derivative of:
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Identical expressions
(x^ three -3x)e^x
(x cubed minus 3x)e to the power of x
(x to the power of three minus 3x)e to the power of x
(x3-3x)ex
x3-3xex
(x³-3x)e^x
(x to the power of 3-3x)e to the power of x
x^3-3xe^x
Similar expressions
(x^3+3x)e^x

The derivative of the function/ (x^3-3x)e^x

Derivative of (x^3-3x)e^x

The solution

You have entered [src]

/ 3      \  x
\x  - 3*x/*e

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

d // 3      \  x\
--\\x  - 3*x/*e /
dx

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{3} - 3 x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{3} - 3 x$ term by term:
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  2. 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: $x$ goes to $1$
      So, the result is: $3$
    So, the result is: $-3$
  The result is: $3 x^{2} - 3$
$g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $e^{x}$ is itself.
The result is: $\left(3 x^{2} - 3\right) e^{x} + \left(x^{3} - 3 x\right) e^{x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/        2\  x   / 3      \  x
\-3 + 3*x /*e  + \x  - 3*x/*e

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of (x^3-3x)e^x