Linear Algebra Examples

$[\begin{array}{c} x & x e^{x} & x^{2} e^{x} \\ 1 & e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 0 & 2 e^{x} + x e^{x} & 2 e^{x} + 4 e^{x} + e^{x} x^{2} \end{array}]$

Step 1

Add $2 e^{x}$ and $4 e^{x}$ .

$[\begin{array}{c} x & x e^{x} & x^{2} e^{x} \\ 1 & e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 0 & 2 e^{x} + x e^{x} & 6 e^{x} + e^{x} x^{2} \end{array}]$

Step 2

Reorder factors in $[\begin{array}{c} x & x e^{x} & x^{2} e^{x} \\ 1 & e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 0 & 2 e^{x} + x e^{x} & 6 e^{x} + e^{x} x^{2} \end{array}]$ .

$[\begin{array}{c} x & x e^{x} & x^{2} e^{x} \\ 1 & e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 0 & 2 e^{x} + x e^{x} & 6 e^{x} + x^{2} e^{x} \end{array}]$

Step 3

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 3.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 3.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 3.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 2 e^{x} + x e^{x} & 6 e^{x} + x^{2} e^{x} \end{array} |$

Step 3.4

Multiply element $a_{11}$ by its cofactor.

$x | \begin{array}{c} e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 2 e^{x} + x e^{x} & 6 e^{x} + x^{2} e^{x} \end{array} |$

Step 3.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} |$

Step 3.6

Multiply element $a_{12}$ by its cofactor.

$- x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} |$

Step 3.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 3.8

Multiply element $a_{13}$ by its cofactor.

$x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 3.9

Add the terms together.

$x | \begin{array}{c} e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 2 e^{x} + x e^{x} & 6 e^{x} + x^{2} e^{x} \end{array} | - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4

Evaluate $| \begin{array}{c} e^{x} + x e^{x} & 2 x e^{x} + x^{2} e^{x} \\ 2 e^{x} + x e^{x} & 6 e^{x} + x^{2} e^{x} \end{array} |$ .

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Step 4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x ((e^{x} + x e^{x}) (6 e^{x} + x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2

Simplify the determinant.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Expand $(e^{x} + x e^{x}) (6 e^{x} + x^{2} e^{x})$ using the FOIL Method.

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Step 4.2.1.1.1

Apply the distributive property.

$x (e^{x} (6 e^{x} + x^{2} e^{x}) + x e^{x} (6 e^{x} + x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.1.2

Apply the distributive property.

$x (e^{x} (6 e^{x}) + e^{x} (x^{2} e^{x}) + x e^{x} (6 e^{x} + x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.1.3

Apply the distributive property.

$x (e^{x} (6 e^{x}) + e^{x} (x^{2} e^{x}) + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Rewrite using the commutative property of multiplication.

$x (6 e^{x} e^{x} + e^{x} (x^{2} e^{x}) + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.2

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.2.2.1

Move $e^{x}$ .

$x (6 (e^{x} e^{x}) + e^{x} (x^{2} e^{x}) + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{x + x} + e^{x} (x^{2} e^{x}) + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.2.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{x} (x^{2} e^{x}) + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.3

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.2.3.1

Move $e^{x}$ .

$x (6 e^{2 x} + e^{x} e^{x} x^{2} + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{x + x} x^{2} + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.3.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + x e^{x} (6 e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.4

Rewrite using the commutative property of multiplication.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{x} e^{x} + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.5

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.2.5.1

Move $e^{x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x (e^{x} e^{x}) + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{x + x} + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.5.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x e^{x} (x^{2} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.2.1.2.6.1

Move $x^{2}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{2} x e^{x} e^{x} - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.6.2

Multiply $x^{2}$ by $x$ .

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Step 4.2.1.2.6.2.1

Raise $x$ to the power of $1$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{2} x^{1} e^{x} e^{x} - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{2 + 1} e^{x} e^{x} - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.6.3

Add $2$ and $1$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{x} e^{x} - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.7

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.2.7.1

Move $e^{x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} (e^{x} e^{x}) - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{x + x} - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.2.7.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - (2 e^{x} + x e^{x}) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.3

Apply the distributive property.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} + (- (2 e^{x}) - (x e^{x})) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.4

Multiply $2$ by $- 1$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} + (- 2 e^{x} - (x e^{x})) (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.5

Expand $(- 2 e^{x} - x e^{x}) (2 x e^{x} + x^{2} e^{x})$ using the FOIL Method.

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Step 4.2.1.5.1

Apply the distributive property.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 2 e^{x} (2 x e^{x} + x^{2} e^{x}) - x e^{x} (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.5.2

Apply the distributive property.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 2 e^{x} (2 x e^{x}) - 2 e^{x} (x^{2} e^{x}) - x e^{x} (2 x e^{x} + x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.5.3

Apply the distributive property.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 2 e^{x} (2 x e^{x}) - 2 e^{x} (x^{2} e^{x}) - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6

Simplify and combine like terms.

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Step 4.2.1.6.1

Simplify each term.

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Step 4.2.1.6.1.1

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.6.1.1.1

Move $e^{x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 2 (e^{x} e^{x}) (2 x) - 2 e^{x} (x^{2} e^{x}) - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 2 e^{x + x} (2 x) - 2 e^{x} (x^{2} e^{x}) - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.1.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 2 e^{2 x} (2 x) - 2 e^{x} (x^{2} e^{x}) - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.2

Rewrite using the commutative property of multiplication.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 2 \cdot 2 e^{2 x} x - 2 e^{x} (x^{2} e^{x}) - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.3

Multiply $- 2$ by $2$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{x} (x^{2} e^{x}) - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.4

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.6.1.4.1

Move $e^{x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 (e^{x} e^{x}) x^{2} - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{x + x} x^{2} - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.4.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - x e^{x} (2 x e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.2.1.6.1.5.1

Move $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - (x \cdot x) e^{x} (2 e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.5.2

Multiply $x$ by $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - x^{2} e^{x} (2 e^{x}) - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.6

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.6.1.6.1

Move $e^{x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - x^{2} (e^{x} e^{x}) \cdot 2 - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.6.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - x^{2} e^{x + x} \cdot 2 - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.6.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - x^{2} e^{2 x} \cdot 2 - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.7

Multiply $2$ by $- 1$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - x e^{x} (x^{2} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.8

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.2.1.6.1.8.1

Move $x^{2}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - (x^{2} x) e^{x} e^{x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.8.2

Multiply $x^{2}$ by $x$ .

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Step 4.2.1.6.1.8.2.1

Raise $x$ to the power of $1$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - (x^{2} x^{1}) e^{x} e^{x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - x^{2 + 1} e^{x} e^{x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.8.3

Add $2$ and $1$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - x^{3} e^{x} e^{x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.9

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 4.2.1.6.1.9.1

Move $e^{x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - x^{3} (e^{x} e^{x})) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.9.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - x^{3} e^{x + x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.1.9.3

Add $x$ and $x$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 e^{2 x} x^{2} - 2 x^{2} e^{2 x} - x^{3} e^{2 x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.2

Subtract $2 x^{2} e^{2 x}$ from $- 2 e^{2 x} x^{2}$ .

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Step 4.2.1.6.2.1

Move $e^{2 x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 2 x^{2} e^{2 x} - 2 x^{2} e^{2 x} - x^{3} e^{2 x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.1.6.2.2

Subtract $2 x^{2} e^{2 x}$ from $- 2 x^{2} e^{2 x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 4 x^{2} e^{2 x} - x^{3} e^{2 x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.2

Combine the opposite terms in $6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} + x^{3} e^{2 x} - 4 e^{2 x} x - 4 x^{2} e^{2 x} - x^{3} e^{2 x}$ .

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Step 4.2.2.1

Subtract $x^{3} e^{2 x}$ from $x^{3} e^{2 x}$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} - 4 e^{2 x} x - 4 x^{2} e^{2 x} + 0) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.2.2

Add $6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} - 4 e^{2 x} x - 4 x^{2} e^{2 x}$ and $0$ .

$x (6 e^{2 x} + e^{2 x} x^{2} + 6 x e^{2 x} - 4 e^{2 x} x - 4 x^{2} e^{2 x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.3

Subtract $4 x^{2} e^{2 x}$ from $e^{2 x} x^{2}$ .

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Step 4.2.3.1

Reorder $e^{2 x}$ and $x^{2}$ .

$x (6 e^{2 x} + x^{2} e^{2 x} - 4 x^{2} e^{2 x} + 6 x e^{2 x} - 4 e^{2 x} x) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.3.2

Subtract $4 x^{2} e^{2 x}$ from $x^{2} e^{2 x}$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 6 x e^{2 x} - 4 e^{2 x} x) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.4

Subtract $4 e^{2 x} x$ from $6 x e^{2 x}$ .

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Step 4.2.4.1

Move $e^{2 x}$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 6 x e^{2 x} - 4 x e^{2 x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 4.2.4.2

Subtract $4 x e^{2 x}$ from $6 x e^{2 x}$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} | \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} | + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 1 & 2 x e^{x} + x^{2} e^{x} \\ 0 & 6 e^{x} + x^{2} e^{x} \end{array} |$ .

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Step 5.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} (1 (6 e^{x} + x^{2} e^{x}) + 0 (2 x e^{x} + x^{2} e^{x})) + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 5.2

Simplify the determinant.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply $6 e^{x} + x^{2} e^{x}$ by $1$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x} + 0 (2 x e^{x} + x^{2} e^{x})) + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 5.2.1.2

Multiply $0$ by $2 x e^{x} + x^{2} e^{x}$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x} + 0) + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 5.2.2

Add $6 e^{x} + x^{2} e^{x}$ and $0$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x}) + x^{2} e^{x} | \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$

Step 6

Evaluate $| \begin{array}{c} 1 & e^{x} + x e^{x} \\ 0 & 2 e^{x} + x e^{x} \end{array} |$ .

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Step 6.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x}) + x^{2} e^{x} (1 (2 e^{x} + x e^{x}) + 0 (e^{x} + x e^{x}))$

Step 6.2

Simplify the determinant.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $2 e^{x} + x e^{x}$ by $1$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x}) + x^{2} e^{x} (2 e^{x} + x e^{x} + 0 (e^{x} + x e^{x}))$

Step 6.2.1.2

Multiply $0$ by $e^{x} + x e^{x}$ .

$x (6 e^{2 x} - 3 x^{2} e^{2 x} + 2 x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x}) + x^{2} e^{x} (2 e^{x} + x e^{x} + 0)$

Step 6.2.2

Add $2 e^{x} + x e^{x}$ and $0$ .

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

Step 7

Simplify the determinant.

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Step 7.1

Simplify each term.

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Simplify.

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Step 7.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 7.1.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.1.2.3

Rewrite using the commutative property of multiplication.

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

Step 7.1.3

Simplify each term.

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Step 7.1.3.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 7.1.3.1.1

Move $x^{2}$ .

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

Step 7.1.3.1.2

Multiply $x^{2}$ by $x$ .

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Step 7.1.3.1.2.1

Raise $x$ to the power of $1$ .

$6 x e^{2 x} - 3 (x^{2} x^{1}) e^{2 x} + 2 x (x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x}) + x^{2} e^{x} (2 e^{x} + x e^{x})$

Step 7.1.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 x e^{2 x} - 3 x^{2 + 1} e^{2 x} + 2 x (x e^{2 x}) - x e^{x} (6 e^{x} + x^{2} e^{x}) + x^{2} e^{x} (2 e^{x} + x e^{x})$

Step 7.1.3.1.3

Add $2$ and $1$ .

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

Step 7.1.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 7.1.3.2.1

Move $x$ .

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

Step 7.1.3.2.2

Multiply $x$ by $x$ .

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

Step 7.1.4

Apply the distributive property.

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

Step 7.1.5

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 7.1.5.1

Move $e^{x}$ .

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

Step 7.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.1.5.3

Add $x$ and $x$ .

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

Step 7.1.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 7.1.6.1

Move $x^{2}$ .

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

Step 7.1.6.2

Multiply $x^{2}$ by $x$ .

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Step 7.1.6.2.1

Raise $x$ to the power of $1$ .

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - x e^{2 x} \cdot 6 - (x^{2} x^{1}) e^{x} e^{x} + x^{2} e^{x} (2 e^{x} + x e^{x})$

Step 7.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - x e^{2 x} \cdot 6 - x^{2 + 1} e^{x} e^{x} + x^{2} e^{x} (2 e^{x} + x e^{x})$

Step 7.1.6.3

Add $2$ and $1$ .

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

Step 7.1.7

Simplify each term.

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Step 7.1.7.1

Multiply $6$ by $- 1$ .

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

Step 7.1.7.2

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 7.1.7.2.1

Move $e^{x}$ .

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

Step 7.1.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.1.7.2.3

Add $x$ and $x$ .

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

Step 7.1.8

Apply the distributive property.

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

Step 7.1.9

Rewrite using the commutative property of multiplication.

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

Step 7.1.10

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 7.1.10.1

Move $x$ .

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{x} e^{x} + x \cdot x^{2} e^{x} e^{x}$

Step 7.1.10.2

Multiply $x$ by $x^{2}$ .

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Step 7.1.10.2.1

Raise $x$ to the power of $1$ .

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{x} e^{x} + x^{1} x^{2} e^{x} e^{x}$

Step 7.1.10.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{x} e^{x} + x^{1 + 2} e^{x} e^{x}$

Step 7.1.10.3

Add $1$ and $2$ .

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{x} e^{x} + x^{3} e^{x} e^{x}$

Step 7.1.11

Simplify each term.

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Step 7.1.11.1

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 7.1.11.1.1

Move $e^{x}$ .

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

Step 7.1.11.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{x + x} + x^{3} e^{x} e^{x}$

Step 7.1.11.1.3

Add $x$ and $x$ .

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{2 x} + x^{3} e^{x} e^{x}$

Step 7.1.11.2

Multiply $e^{x}$ by $e^{x}$ by adding the exponents.

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Step 7.1.11.2.1

Move $e^{x}$ .

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

Step 7.1.11.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{2 x} + x^{3} e^{x + x}$

Step 7.1.11.2.3

Add $x$ and $x$ .

$6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{2 x} + x^{3} e^{2 x}$

Step 7.2

Combine the opposite terms in $6 x e^{2 x} - 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} - 6 x e^{2 x} - x^{3} e^{2 x} + 2 x^{2} e^{2 x} + x^{3} e^{2 x}$ .

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Step 7.2.1

Subtract $6 x e^{2 x}$ from $6 x e^{2 x}$ .

$- 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} + 0 - x^{3} e^{2 x} + 2 x^{2} e^{2 x} + x^{3} e^{2 x}$

Step 7.2.2

Add $- 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x}$ and $0$ .

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

Step 7.2.3

Add $- x^{3} e^{2 x}$ and $x^{3} e^{2 x}$ .

$- 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} + 2 x^{2} e^{2 x} + 0$

Step 7.2.4

Add $- 3 x^{3} e^{2 x} + 2 x^{2} e^{2 x} + 2 x^{2} e^{2 x}$ and $0$ .

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

Step 7.3

Add $2 x^{2} e^{2 x}$ and $2 x^{2} e^{2 x}$ .

$- 3 x^{3} e^{2 x} + 4 x^{2} e^{2 x}$