Calculus Examples

$f (x) = \frac{x d}{d x} \cdot (x + 1) + \frac{(x + 1) d}{d x} \cdot e^{x^{6}}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $\frac{x d}{d x} \cdot (x + 1) + \frac{(x + 1) d}{d x} \cdot e^{x^{6}}$ with respect to $x$ is $\frac{d}{d x} [\frac{x d}{d x} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$ .

$\frac{d}{d x} [\frac{x d}{d x} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{x d}{d x} \cdot (x + 1)]$ .

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Step 1.2.1

Cancel the common factor of $x$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{d}{d x} [\frac{x d}{d x} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.1.2

Rewrite the expression.

$\frac{d}{d x} [\frac{d}{d} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.2

Cancel the common factor of $d$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{d}{d x} [\frac{d}{d} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.2.2

Rewrite the expression.

$\frac{d}{d x} [1 \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.3

Multiply $x + 1$ by $1$ .

$\frac{d}{d x} [x + 1] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.4

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0 + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.2.7

Add $1$ and $0$ .

$1 + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$ .

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Step 1.3.1

Cancel the common factor of $d$ .

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Step 1.3.1.1

Cancel the common factor.

$1 + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 1.3.1.2

Rewrite the expression.

$1 + \frac{d}{d x} [\frac{x + 1}{x} \cdot e^{x^{6}}]$

Step 1.3.2

Combine $\frac{x + 1}{x}$ and $e^{x^{6}}$ .

$1 + \frac{d}{d x} [\frac{(x + 1) e^{x^{6}}}{x}]$

Step 1.3.3

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = (x + 1) e^{x^{6}}$ and $g (x) = x$ .

$1 + \frac{x \frac{d}{d x} [(x + 1) e^{x^{6}}] - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 1$ and $g (x) = e^{x^{6}}$ .

$1 + \frac{x ((x + 1) \frac{d}{d x} [e^{x^{6}}] + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{6}$ .

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Step 1.3.5.1

To apply the Chain Rule, set $u$ as $x^{6}$ .

$1 + \frac{x ((x + 1) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{6}]) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.5.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$1 + \frac{x ((x + 1) (e^{u} \frac{d}{d x} [x^{6}]) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.5.3

Replace all occurrences of $u$ with $x^{6}$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} \frac{d}{d x} [x^{6}]) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.7

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (\frac{d}{d x} [x] + \frac{d}{d x} [1])) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (1 + \frac{d}{d x} [1])) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.9

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (1 + 0)) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 1.3.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (1 + 0)) - (x + 1) e^{x^{6}} \cdot 1}{x^{2}}$

Step 1.3.11

Add $1$ and $0$ .

$1 + \frac{x ((x + 1) e^{x^{6}} (6 x^{5}) + e^{x^{6}} \cdot 1) - (x + 1) e^{x^{6}} \cdot 1}{x^{2}}$

Step 1.3.12

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

$1 + \frac{x ((x + 1) e^{x^{6}} (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}} \cdot 1}{x^{2}}$

Step 1.3.13

Multiply $- 1$ by $1$ .

$1 + \frac{x ((x + 1) e^{x^{6}} (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}}}{x^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$1 + \frac{x ((x e^{x^{6}} + 1 e^{x^{6}}) (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}}}{x^{2}}$

Step 1.4.2

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5}) + 1 e^{x^{6}} (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}}}{x^{2}}$

Step 1.4.3

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - (x + 1) e^{x^{6}}}{x^{2}}$

Step 1.4.4

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} + (- x - 1 \cdot 1) e^{x^{6}}}{x^{2}}$

Step 1.4.5

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6

Combine terms.

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Step 1.4.6.1

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

$1 + \frac{x (x^{5} x^{1} e^{x^{6}} \cdot 6) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.2

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

$1 + \frac{x (x^{5 + 1} e^{x^{6}} \cdot 6) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.3

Add $5$ and $1$ .

$1 + \frac{x (x^{6} e^{x^{6}} \cdot 6) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.4

Move $6$ to the left of $x^{6} e^{x^{6}}$ .

$1 + \frac{x (6 \cdot (x^{6} e^{x^{6}})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.5

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

$1 + \frac{6 (x^{1} x^{6}) e^{x^{6}} + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.6

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

$1 + \frac{6 x^{1 + 6} e^{x^{6}} + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.7

Add $1$ and $6$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.8

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

$1 + \frac{6 x^{7} e^{x^{6}} + x (e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.9

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

$1 + \frac{6 x^{7} e^{x^{6}} + x^{5} x^{1} (e^{x^{6}} \cdot (6)) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.10

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

$1 + \frac{6 x^{7} e^{x^{6}} + x^{5 + 1} (e^{x^{6}} \cdot (6)) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.11

Add $5$ and $1$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (e^{x^{6}} \cdot (6)) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.12

Move $6$ to the left of $e^{x^{6}}$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 \cdot e^{x^{6}}) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.13

Multiply $- 1$ by $1$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) + x e^{x^{6}} - x e^{x^{6}} - 1 e^{x^{6}}}{x^{2}}$

Step 1.4.6.14

Rewrite $- 1 e^{x^{6}}$ as $- e^{x^{6}}$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) + x e^{x^{6}} - x e^{x^{6}} - e^{x^{6}}}{x^{2}}$

Step 1.4.6.15

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

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) + 0 - e^{x^{6}}}{x^{2}}$

Step 1.4.6.16

Add $6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}})$ and $0$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) - e^{x^{6}}}{x^{2}}$

Step 1.4.6.17

Write $1$ as a fraction with a common denominator.

$\frac{x^{2}}{x^{2}} + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) - e^{x^{6}}}{x^{2}}$

Step 1.4.6.18

Combine the numerators over the common denominator.

$\frac{x^{2} + 6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) - e^{x^{6}}}{x^{2}}$

Step 1.4.7

Reorder terms.

$\frac{x^{2} + 6 e^{x^{6}} x^{7} + 6 e^{x^{6}} x^{6} - e^{x^{6}}}{x^{2}}$

Step 1.4.8

Reorder factors in $\frac{x^{2} + 6 e^{x^{6}} x^{7} + 6 e^{x^{6}} x^{6} - e^{x^{6}}}{x^{2}}$ .

$\frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

$f'' (x) = \frac{x^{2} \frac{d}{d x} (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{{(x^{2})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = \frac{x^{2} \frac{d}{d x} (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{2 \cdot 2}}$

Step 2.2.1.2

Multiply $2$ by $2$ .

$f'' (x) = \frac{x^{2} \frac{d}{d x} (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.2

By the Sum Rule, the derivative of $x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [6 x^{7} e^{x^{6}}] + \frac{d}{d x} [6 x^{6} e^{x^{6}}] + \frac{d}{d x} [- e^{x^{6}}]$ .

$f'' (x) = \frac{x^{2} (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (6 x^{7} e^{x^{6}}) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = \frac{x^{2} (2 x + \frac{d}{d x} (6 x^{7} e^{x^{6}}) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.2.4

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{7} e^{x^{6}}$ with respect to $x$ is $6 \frac{d}{d x} [x^{7} e^{x^{6}}]$ .

$f'' (x) = \frac{x^{2} (2 x + 6 \frac{d}{d x} (x^{7} e^{x^{6}}) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{7}$ and $g (x) = e^{x^{6}}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{7} \frac{d}{d x} (e^{x^{6}}) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{6}$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{1}$ as $x^{6}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{7} (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (x^{6})) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{7} (e^{u_{1}} \frac{d}{d x} (x^{6})) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.4.3

Replace all occurrences of $u_{1}$ with $x^{6}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{7} (e^{x^{6}} \frac{d}{d x} (x^{6})) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{7} (e^{x^{6}} (6 x^{5})) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.6

Multiply $x^{7}$ by $x^{5}$ by adding the exponents.

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Step 2.6.1

Move $x^{5}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{5} x^{7} (e^{x^{6}} \cdot (6)) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.6.2

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

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{5 + 7} (e^{x^{6}} \cdot (6)) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.6.3

Add $5$ and $7$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (e^{x^{6}} \cdot (6)) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.7

Differentiate.

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Step 2.7.1

Move $6$ to the left of $e^{x^{6}}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 \cdot e^{x^{6}}) + e^{x^{6}} \frac{d}{d x} (x^{7})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.7.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + \frac{d}{d x} (6 x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.7.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{6} e^{x^{6}}$ with respect to $x$ is $6 \frac{d}{d x} [x^{6} e^{x^{6}}]$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 \frac{d}{d x} (x^{6} e^{x^{6}}) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.8

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{6}$ and $g (x) = e^{x^{6}}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{6} \frac{d}{d x} (e^{x^{6}}) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{6}$ .

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Step 2.9.1

To apply the Chain Rule, set $u_{2}$ as $x^{6}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{6} (\frac{d}{d u (2)} (e^{u_{2}}) \frac{d}{d x} (x^{6})) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.9.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{6} (e^{u_{2}} \frac{d}{d x} (x^{6})) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.9.3

Replace all occurrences of $u_{2}$ with $x^{6}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{6} (e^{x^{6}} \frac{d}{d x} (x^{6})) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{6} (e^{x^{6}} (6 x^{5})) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.11

Multiply $x^{6}$ by $x^{5}$ by adding the exponents.

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Step 2.11.1

Move $x^{5}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{5} x^{6} (e^{x^{6}} \cdot (6)) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.11.2

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

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{5 + 6} (e^{x^{6}} \cdot (6)) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.11.3

Add $5$ and $6$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (e^{x^{6}} \cdot (6)) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.12

Differentiate.

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Step 2.12.1

Move $6$ to the left of $e^{x^{6}}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 \cdot e^{x^{6}}) + e^{x^{6}} \frac{d}{d x} (x^{6})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.12.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) + \frac{d}{d x} (- e^{x^{6}})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.12.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{x^{6}}$ with respect to $x$ is $- \frac{d}{d x} [e^{x^{6}}]$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - \frac{d}{d x} e^{x^{6}}) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.13

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{6}$ .

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Step 2.13.1

To apply the Chain Rule, set $u_{3}$ as $x^{6}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - (\frac{d}{d u (3)} (e^{u_{3}}) \frac{d}{d x} (x^{6}))) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.13.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - (e^{u_{3}} \frac{d}{d x} (x^{6}))) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.13.3

Replace all occurrences of $u_{3}$ with $x^{6}$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - (e^{x^{6}} \frac{d}{d x} (x^{6}))) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.14

Differentiate using the Power Rule.

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Step 2.14.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - e^{x^{6}} (6 x^{5})) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.14.2

Multiply $6$ by $- 1$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5}) - (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) \frac{d}{d x} x^{2}}{x^{4}}$

Step 2.14.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

Step 2.14.4

Simplify with factoring out.

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Step 2.14.4.1

Multiply $2$ by $- 1$ .

$f'' (x) = \frac{x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5}) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) x}{x^{4}}$

Step 2.14.4.2

Factor $x$ out of $x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5}) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) x$ .

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Step 2.14.4.2.1

Factor $x$ out of $x^{2} (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5})$ .

$f'' (x) = \frac{x (x (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5})) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) x}{x^{4}}$

Step 2.14.4.2.2

Factor $x$ out of $- 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}) x$ .

$f'' (x) = \frac{x (x (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5})) + x (- 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}))}{x^{4}}$

Step 2.14.4.2.3

Factor $x$ out of $x (x (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5})) + x (- 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}))$ .

$f'' (x) = \frac{x (x (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5}) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}))}{x^{4}}$

Step 2.15

Cancel the common factors.

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Step 2.15.1

Factor $x$ out of $x^{4}$ .

Step 2.15.2

Cancel the common factor.

Step 2.15.3

Rewrite the expression.

$f'' (x) = \frac{x (2 x + 6 (x^{12} (6 e^{x^{6}}) + e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5}) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}})}{x^{3}}$

Step 2.16

Simplify.

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Step 2.16.1

Apply the distributive property.

$f'' (x) = \frac{x (2 x + 6 (x^{12} (6 e^{x^{6}})) + 6 (e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}}) + e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5}) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}})}{x^{3}}$

Step 2.16.2

Apply the distributive property.

$f'' (x) = \frac{x (2 x + 6 (x^{12} (6 e^{x^{6}})) + 6 (e^{x^{6}} (7 x^{6})) + 6 (x^{11} (6 e^{x^{6}})) + 6 (e^{x^{6}} (6 x^{5})) - 6 e^{x^{6}} x^{5}) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}})}{x^{3}}$

Step 2.16.3

Apply the distributive property.

$f'' (x) = \frac{x (2 x) + x (6 (x^{12} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 (x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}})}{x^{3}}$

Step 2.16.4

Apply the distributive property.

$f'' (x) = \frac{x (2 x) + x (6 (x^{12} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5

Simplify the numerator.

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Step 2.16.5.1

Simplify each term.

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Step 2.16.5.1.1

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x \cdot x + x (6 (x^{12} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.2

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

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Step 2.16.5.1.2.1

Move $x$ .

$f'' (x) = \frac{2 (x \cdot x) + x (6 (x^{12} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.2.2

Multiply $x$ by $x$ .

$f'' (x) = \frac{2 x^{2} + x (6 (x^{12} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.3

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x^{2} + 6 \cdot (6 x (x^{12} e^{x^{6}})) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.4

Multiply $x$ by $x^{12}$ by adding the exponents.

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Step 2.16.5.1.4.1

Move $x^{12}$ .

$f'' (x) = \frac{2 x^{2} + 6 \cdot (6 (x^{12} x) e^{x^{6}}) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.4.2

Multiply $x^{12}$ by $x$ .

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Step 2.16.5.1.4.2.1

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

Step 2.16.5.1.4.2.2

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

$f'' (x) = \frac{2 x^{2} + 6 \cdot (6 x^{12 + 1} e^{x^{6}}) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.4.3

Add $12$ and $1$ .

$f'' (x) = \frac{2 x^{2} + 6 \cdot (6 x^{13} e^{x^{6}}) + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.5

Multiply $6$ by $6$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + x (6 (e^{x^{6}} (7 x^{6}))) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.6

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 6 \cdot (7 x (e^{x^{6}} x^{6})) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.7

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 2.16.5.1.7.1

Move $x^{6}$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 6 \cdot (7 (x^{6} x) e^{x^{6}}) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.7.2

Multiply $x^{6}$ by $x$ .

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Step 2.16.5.1.7.2.1

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

Step 2.16.5.1.7.2.2

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

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 6 \cdot (7 x^{6 + 1} e^{x^{6}}) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.7.3

Add $6$ and $1$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 6 \cdot (7 x^{7} e^{x^{6}}) + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.8

Multiply $6$ by $7$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + x (6 (x^{11} (6 e^{x^{6}}))) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.9

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 6 \cdot (6 x (x^{11} e^{x^{6}})) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.10

Multiply $x$ by $x^{11}$ by adding the exponents.

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Step 2.16.5.1.10.1

Move $x^{11}$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 6 \cdot (6 (x^{11} x) e^{x^{6}}) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.10.2

Multiply $x^{11}$ by $x$ .

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Step 2.16.5.1.10.2.1

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

Step 2.16.5.1.10.2.2

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

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 6 \cdot (6 x^{11 + 1} e^{x^{6}}) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.10.3

Add $11$ and $1$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 6 \cdot (6 x^{12} e^{x^{6}}) + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.11

Multiply $6$ by $6$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + x (6 (e^{x^{6}} (6 x^{5}))) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.12

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 6 \cdot (6 x (e^{x^{6}} x^{5})) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.13

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 2.16.5.1.13.1

Move $x^{5}$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 6 \cdot (6 (x^{5} x) e^{x^{6}}) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.13.2

Multiply $x^{5}$ by $x$ .

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Step 2.16.5.1.13.2.1

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

Step 2.16.5.1.13.2.2

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

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 6 \cdot (6 x^{5 + 1} e^{x^{6}}) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.13.3

Add $5$ and $1$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 6 \cdot (6 x^{6} e^{x^{6}}) + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.14

Multiply $6$ by $6$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} + x (- 6 e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.15

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x (e^{x^{6}} x^{5}) - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.16

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 2.16.5.1.16.1

Move $x^{5}$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 (x^{5} x) e^{x^{6}} - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.16.2

Multiply $x^{5}$ by $x$ .

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Step 2.16.5.1.16.2.1

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

Step 2.16.5.1.16.2.2

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

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{5 + 1} e^{x^{6}} - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.16.3

Add $5$ and $1$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} - 2 x^{2} - 2 (6 x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.17

Multiply $6$ by $- 2$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} - 2 x^{2} - 12 (x^{7} e^{x^{6}}) - 2 (6 x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.18

Multiply $6$ by $- 2$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} - 2 x^{2} - 12 x^{7} e^{x^{6}} - 12 (x^{6} e^{x^{6}}) - 2 (- e^{x^{6}})}{x^{3}}$

Step 2.16.5.1.19

Multiply $- 1$ by $- 2$ .

$f'' (x) = \frac{2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} - 2 x^{2} - 12 x^{7} e^{x^{6}} - 12 x^{6} e^{x^{6}} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.5.2

Combine the opposite terms in $2 x^{2} + 36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} - 2 x^{2} - 12 x^{7} e^{x^{6}} - 12 x^{6} e^{x^{6}} + 2 e^{x^{6}}$ .

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Step 2.16.5.2.1

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

$f'' (x) = \frac{36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} + 0 - 12 x^{7} e^{x^{6}} - 12 x^{6} e^{x^{6}} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.5.2.2

Add $36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}}$ and $0$ .

$f'' (x) = \frac{36 x^{13} e^{x^{6}} + 42 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} - 12 x^{7} e^{x^{6}} - 12 x^{6} e^{x^{6}} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.5.3

Subtract $12 x^{7} e^{x^{6}}$ from $42 x^{7} e^{x^{6}}$ .

$f'' (x) = \frac{36 x^{13} e^{x^{6}} + 30 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 36 x^{6} e^{x^{6}} - 6 x^{6} e^{x^{6}} - 12 x^{6} e^{x^{6}} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.5.4

Subtract $6 x^{6} e^{x^{6}}$ from $36 x^{6} e^{x^{6}}$ .

$f'' (x) = \frac{36 x^{13} e^{x^{6}} + 30 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 30 x^{6} e^{x^{6}} - 12 x^{6} e^{x^{6}} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.5.5

Subtract $12 x^{6} e^{x^{6}}$ from $30 x^{6} e^{x^{6}}$ .

$f'' (x) = \frac{36 x^{13} e^{x^{6}} + 30 x^{7} e^{x^{6}} + 36 x^{12} e^{x^{6}} + 18 x^{6} e^{x^{6}} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.6

Reorder terms.

$f'' (x) = \frac{36 e^{x^{6}} x^{13} + 30 e^{x^{6}} x^{7} + 36 e^{x^{6}} x^{12} + 18 e^{x^{6}} x^{6} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.7

Simplify the numerator.

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Step 2.16.7.1

Factor $2 e^{x^{6}}$ out of $36 e^{x^{6}} x^{13} + 30 e^{x^{6}} x^{7} + 36 e^{x^{6}} x^{12} + 18 e^{x^{6}} x^{6} + 2 e^{x^{6}}$ .

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Step 2.16.7.1.1

Factor $2 e^{x^{6}}$ out of $36 e^{x^{6}} x^{13}$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13}) + 30 e^{x^{6}} x^{7} + 36 e^{x^{6}} x^{12} + 18 e^{x^{6}} x^{6} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.7.1.2

Factor $2 e^{x^{6}}$ out of $30 e^{x^{6}} x^{7}$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13}) + 2 e^{x^{6}} (15 x^{7}) + 36 e^{x^{6}} x^{12} + 18 e^{x^{6}} x^{6} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.7.1.3

Factor $2 e^{x^{6}}$ out of $36 e^{x^{6}} x^{12}$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13}) + 2 e^{x^{6}} (15 x^{7}) + 2 e^{x^{6}} (18 x^{12}) + 18 e^{x^{6}} x^{6} + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.7.1.4

Factor $2 e^{x^{6}}$ out of $18 e^{x^{6}} x^{6}$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13}) + 2 e^{x^{6}} (15 x^{7}) + 2 e^{x^{6}} (18 x^{12}) + 2 e^{x^{6}} (9 x^{6}) + 2 e^{x^{6}}}{x^{3}}$

Step 2.16.7.1.5

Factor $2 e^{x^{6}}$ out of $2 e^{x^{6}}$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13}) + 2 e^{x^{6}} (15 x^{7}) + 2 e^{x^{6}} (18 x^{12}) + 2 e^{x^{6}} (9 x^{6}) + 2 e^{x^{6}} (1)}{x^{3}}$

Step 2.16.7.1.6

Factor $2 e^{x^{6}}$ out of $2 e^{x^{6}} (18 x^{13}) + 2 e^{x^{6}} (15 x^{7})$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13} + 15 x^{7}) + 2 e^{x^{6}} (18 x^{12}) + 2 e^{x^{6}} (9 x^{6}) + 2 e^{x^{6}} (1)}{x^{3}}$

Step 2.16.7.1.7

Factor $2 e^{x^{6}}$ out of $2 e^{x^{6}} (18 x^{13} + 15 x^{7}) + 2 e^{x^{6}} (18 x^{12})$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13} + 15 x^{7} + 18 x^{12}) + 2 e^{x^{6}} (9 x^{6}) + 2 e^{x^{6}} (1)}{x^{3}}$

Step 2.16.7.1.8

Factor $2 e^{x^{6}}$ out of $2 e^{x^{6}} (18 x^{13} + 15 x^{7} + 18 x^{12}) + 2 e^{x^{6}} (9 x^{6})$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13} + 15 x^{7} + 18 x^{12} + 9 x^{6}) + 2 e^{x^{6}} (1)}{x^{3}}$

Step 2.16.7.1.9

Factor $2 e^{x^{6}}$ out of $2 e^{x^{6}} (18 x^{13} + 15 x^{7} + 18 x^{12} + 9 x^{6}) + 2 e^{x^{6}} (1)$ .

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13} + 15 x^{7} + 18 x^{12} + 9 x^{6} + 1)}{x^{3}}$

Step 2.16.7.2

Reorder terms.

$f'' (x) = \frac{2 e^{x^{6}} (18 x^{13} + 18 x^{12} + 15 x^{7} + 9 x^{6} + 1)}{x^{3}}$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$\frac{d}{d x} [\frac{x d}{d x} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{x d}{d x} \cdot (x + 1)]$ .

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Step 4.1.2.1

Cancel the common factor of $x$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{d}{d x} [\frac{x d}{d x} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.1.2

Rewrite the expression.

$\frac{d}{d x} [\frac{d}{d} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.2

Cancel the common factor of $d$ .

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Step 4.1.2.2.1

Cancel the common factor.

$\frac{d}{d x} [\frac{d}{d} \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.2.2

Rewrite the expression.

$\frac{d}{d x} [1 \cdot (x + 1)] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.3

Multiply $x + 1$ by $1$ .

$\frac{d}{d x} [x + 1] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.4

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1] + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0 + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.2.7

Add $1$ and $0$ .

$1 + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$ .

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Step 4.1.3.1

Cancel the common factor of $d$ .

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Step 4.1.3.1.1

Cancel the common factor.

$1 + \frac{d}{d x} [\frac{(x + 1) d}{d x} \cdot e^{x^{6}}]$

Step 4.1.3.1.2

Rewrite the expression.

$1 + \frac{d}{d x} [\frac{x + 1}{x} \cdot e^{x^{6}}]$

Step 4.1.3.2

Combine $\frac{x + 1}{x}$ and $e^{x^{6}}$ .

$1 + \frac{d}{d x} [\frac{(x + 1) e^{x^{6}}}{x}]$

Step 4.1.3.3

$1 + \frac{x \frac{d}{d x} [(x + 1) e^{x^{6}}] - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x + 1$ and $g (x) = e^{x^{6}}$ .

$1 + \frac{x ((x + 1) \frac{d}{d x} [e^{x^{6}}] + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x^{6}$ .

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Step 4.1.3.5.1

To apply the Chain Rule, set $u$ as $x^{6}$ .

$1 + \frac{x ((x + 1) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x^{6}]) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.5.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$1 + \frac{x ((x + 1) (e^{u} \frac{d}{d x} [x^{6}]) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.5.3

Replace all occurrences of $u$ with $x^{6}$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} \frac{d}{d x} [x^{6}]) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} \frac{d}{d x} [x + 1]) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.7

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (\frac{d}{d x} [x] + \frac{d}{d x} [1])) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (1 + \frac{d}{d x} [1])) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.9

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (1 + 0)) - (x + 1) e^{x^{6}} \frac{d}{d x} [x]}{x^{2}}$

Step 4.1.3.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{x ((x + 1) (e^{x^{6}} (6 x^{5})) + e^{x^{6}} (1 + 0)) - (x + 1) e^{x^{6}} \cdot 1}{x^{2}}$

Step 4.1.3.11

Add $1$ and $0$ .

$1 + \frac{x ((x + 1) e^{x^{6}} (6 x^{5}) + e^{x^{6}} \cdot 1) - (x + 1) e^{x^{6}} \cdot 1}{x^{2}}$

Step 4.1.3.12

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

$1 + \frac{x ((x + 1) e^{x^{6}} (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}} \cdot 1}{x^{2}}$

Step 4.1.3.13

Multiply $- 1$ by $1$ .

$1 + \frac{x ((x + 1) e^{x^{6}} (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}}}{x^{2}}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Apply the distributive property.

$1 + \frac{x ((x e^{x^{6}} + 1 e^{x^{6}}) (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}}}{x^{2}}$

Step 4.1.4.2

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5}) + 1 e^{x^{6}} (6 x^{5}) + e^{x^{6}}) - (x + 1) e^{x^{6}}}{x^{2}}$

Step 4.1.4.3

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - (x + 1) e^{x^{6}}}{x^{2}}$

Step 4.1.4.4

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} + (- x - 1 \cdot 1) e^{x^{6}}}{x^{2}}$

Step 4.1.4.5

Apply the distributive property.

$1 + \frac{x (x e^{x^{6}} (6 x^{5})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6

Combine terms.

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Step 4.1.4.6.1

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

$1 + \frac{x (x^{5} x^{1} e^{x^{6}} \cdot 6) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.2

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

$1 + \frac{x (x^{5 + 1} e^{x^{6}} \cdot 6) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.3

Add $5$ and $1$ .

$1 + \frac{x (x^{6} e^{x^{6}} \cdot 6) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.4

Move $6$ to the left of $x^{6} e^{x^{6}}$ .

$1 + \frac{x (6 \cdot (x^{6} e^{x^{6}})) + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.5

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

$1 + \frac{6 (x^{1} x^{6}) e^{x^{6}} + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.6

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

$1 + \frac{6 x^{1 + 6} e^{x^{6}} + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.7

Add $1$ and $6$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x (1 e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.8

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

$1 + \frac{6 x^{7} e^{x^{6}} + x (e^{x^{6}} (6 x^{5})) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.9

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

$1 + \frac{6 x^{7} e^{x^{6}} + x^{5} x^{1} (e^{x^{6}} \cdot (6)) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.10

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

$1 + \frac{6 x^{7} e^{x^{6}} + x^{5 + 1} (e^{x^{6}} \cdot (6)) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.11

Add $5$ and $1$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (e^{x^{6}} \cdot (6)) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.12

Move $6$ to the left of $e^{x^{6}}$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 \cdot e^{x^{6}}) + x e^{x^{6}} - x e^{x^{6}} - 1 \cdot 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.13

Multiply $- 1$ by $1$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) + x e^{x^{6}} - x e^{x^{6}} - 1 e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.14

Rewrite $- 1 e^{x^{6}}$ as $- e^{x^{6}}$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) + x e^{x^{6}} - x e^{x^{6}} - e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.15

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

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) + 0 - e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.16

Add $6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}})$ and $0$ .

$1 + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) - e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.17

Write $1$ as a fraction with a common denominator.

$\frac{x^{2}}{x^{2}} + \frac{6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) - e^{x^{6}}}{x^{2}}$

Step 4.1.4.6.18

Combine the numerators over the common denominator.

$\frac{x^{2} + 6 x^{7} e^{x^{6}} + x^{6} (6 e^{x^{6}}) - e^{x^{6}}}{x^{2}}$

Step 4.1.4.7

Reorder terms.

$\frac{x^{2} + 6 e^{x^{6}} x^{7} + 6 e^{x^{6}} x^{6} - e^{x^{6}}}{x^{2}}$

Step 4.1.4.8

Reorder factors in $\frac{x^{2} + 6 e^{x^{6}} x^{7} + 6 e^{x^{6}} x^{6} - e^{x^{6}}}{x^{2}}$ .

$f' (x) = \frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}}$ .

$\frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}} = 0$

Step 5.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.63217197$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{x^{2} + 6 x^{7} e^{x^{6}} + 6 x^{6} e^{x^{6}} - e^{x^{6}}}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = 0.63217197$

Step 8

Evaluate the second derivative at $x = 0.63217197$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2 e^{{(0.63217197)}^{6}} (18 {(0.63217197)}^{13} + 18 {(0.63217197)}^{12} + 15 {(0.63217197)}^{7} + 9 {(0.63217197)}^{6} + 1)}{{(0.63217197)}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $0.63217197$ to the power of $13$ .

$\frac{2 e^{{0.63217197}^{6}} (18 \cdot 0.00257547 + 18 \cdot {0.63217197}^{12} + 15 \cdot {0.63217197}^{7} + 9 \cdot {0.63217197}^{6} + 1)}{{0.63217197}^{3}}$

Step 9.1.2

Multiply $18$ by $0.00257547$ .

$\frac{2 e^{{0.63217197}^{6}} (0.04635862 + 18 \cdot {0.63217197}^{12} + 15 \cdot {0.63217197}^{7} + 9 \cdot {0.63217197}^{6} + 1)}{{0.63217197}^{3}}$

Step 9.1.3

Raise $0.63217197$ to the power of $12$ .

$\frac{2 e^{{0.63217197}^{6}} (0.04635862 + 18 \cdot 0.00407401 + 15 \cdot {0.63217197}^{7} + 9 \cdot {0.63217197}^{6} + 1)}{{0.63217197}^{3}}$

Step 9.1.4

Multiply $18$ by $0.00407401$ .

$\frac{2 e^{{0.63217197}^{6}} (0.04635862 + 0.0733323 + 15 \cdot {0.63217197}^{7} + 9 \cdot {0.63217197}^{6} + 1)}{{0.63217197}^{3}}$

Step 9.1.5

Raise $0.63217197$ to the power of $7$ .

$\frac{2 e^{{0.63217197}^{6}} (0.04635862 + 0.0733323 + 15 \cdot 0.04035028 + 9 \cdot {0.63217197}^{6} + 1)}{{0.63217197}^{3}}$

Step 9.1.6

Multiply $15$ by $0.04035028$ .

$\frac{2 e^{{0.63217197}^{6}} (0.04635862 + 0.0733323 + 0.60525434 + 9 \cdot {0.63217197}^{6} + 1)}{{0.63217197}^{3}}$

Step 9.1.7

Raise $0.63217197$ to the power of $6$ .

$\frac{2 e^{{0.63217197}^{6}} (0.04635862 + 0.0733323 + 0.60525434 + 9 \cdot 0.06382802 + 1)}{{0.63217197}^{3}}$

Step 9.1.8

Multiply $9$ by $0.06382802$ .

$\frac{2 e^{{0.63217197}^{6}} (0.04635862 + 0.0733323 + 0.60525434 + 0.57445223 + 1)}{{0.63217197}^{3}}$

Step 9.1.9

Add $0.04635862$ and $0.0733323$ .

$\frac{2 e^{{0.63217197}^{6}} (0.11969093 + 0.60525434 + 0.57445223 + 1)}{{0.63217197}^{3}}$

Step 9.1.10

Add $0.11969093$ and $0.60525434$ .

$\frac{2 e^{{0.63217197}^{6}} (0.72494527 + 0.57445223 + 1)}{{0.63217197}^{3}}$

Step 9.1.11

Add $0.72494527$ and $0.57445223$ .

$\frac{2 e^{{0.63217197}^{6}} (1.29939751 + 1)}{{0.63217197}^{3}}$

Step 9.1.12

Add $1.29939751$ and $1$ .

$\frac{2 e^{{0.63217197}^{6}} \cdot 2.29939751}{{0.63217197}^{3}}$

Step 9.1.13

Combine exponents.

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Step 9.1.13.1

Multiply $2.29939751$ by $2$ .

$\frac{4.59879502 e^{{0.63217197}^{6}}}{{0.63217197}^{3}}$

Step 9.1.13.2

Multiply $4.59879502$ by $e^{{0.63217197}^{6}}$ .

$\frac{4.90189735}{{0.63217197}^{3}}$

Step 9.2

Simplify the expression.

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Step 9.2.1

Raise $0.63217197$ to the power of $3$ .

$\frac{4.90189735}{0.25264209}$

Step 9.2.2

Divide $4.90189735$ by $0.25264209$ .

$19.40253627$

Step 10

$x = 0.63217197$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0.63217197$ is a local minimum

Step 11

Find the y-value when $x = 0.63217197$ .

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Step 11.1

Replace the variable $x$ with $0.63217197$ in the expression.

$f (0.63217197) = (0.63217197) \cdot \frac{d}{d (0.63217197)} \cdot ((0.63217197) + 1) + ((0.63217197) + 1) \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Cancel the common factor of $d$ .

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Step 11.2.1.1.1

Cancel the common factor.

$f (0.63217197) = 0.63217197 \cdot \frac{d}{d \cdot 0.63217197} \cdot ((0.63217197) + 1) + ((0.63217197) + 1) \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.1.2

Rewrite the expression.

$f (0.63217197) = 0.63217197 \cdot \frac{1}{0.63217197} \cdot ((0.63217197) + 1) + ((0.63217197) + 1) \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.2

Divide $1$ by $0.63217197$ .

$f (0.63217197) = 0.63217197 \cdot 1.58184805 \cdot ((0.63217197) + 1) + ((0.63217197) + 1) \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.3

Multiply $0.63217197$ by $1.58184805$ .

$f (0.63217197) = 1 \cdot ((0.63217197) + 1) + ((0.63217197) + 1) \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.4

Multiply $(0.63217197) + 1$ by $1$ .

$f (0.63217197) = (0.63217197) + 1 + ((0.63217197) + 1) \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.5

Add $0.63217197$ and $1$ .

$f (0.63217197) = 1.63217197 + ((0.63217197) + 1) \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.6

Add $0.63217197$ and $1$ .

$f (0.63217197) = 1.63217197 + 1.63217197 \cdot \frac{d}{d (0.63217197)} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.7

Cancel the common factor of $d$ .

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Step 11.2.1.7.1

Cancel the common factor.

$f (0.63217197) = 1.63217197 + 1.63217197 \cdot \frac{d}{d \cdot 0.63217197} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.7.2

Rewrite the expression.

$f (0.63217197) = 1.63217197 + 1.63217197 \cdot \frac{1}{0.63217197} \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.8

Divide $1$ by $0.63217197$ .

$f (0.63217197) = 1.63217197 + 1.63217197 \cdot 1.58184805 \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.9

Multiply $1.63217197$ by $1.58184805$ .

$f (0.63217197) = 1.63217197 + 2.58184805 \cdot e^{{(0.63217197)}^{6}}$

Step 11.2.1.10

Raise $0.63217197$ to the power of $6$ .

$f (0.63217197) = 1.63217197 + 2.58184805 \cdot e^{0.06382802}$

Step 11.2.1.11

Multiply $2.58184805$ by $e^{0.06382802}$ .

$f (0.63217197) = 1.63217197 + 2.75201526$

Step 11.2.2

Add $1.63217197$ and $2.75201526$ .

$f (0.63217197) = 4.38418723$

Step 11.2.3

The final answer is $4.38418723$ .

$y = 4.38418723$

Step 12

These are the local extrema for $f (x) = \frac{x d}{d x} \cdot (x + 1) + \frac{(x + 1) d}{d x} \cdot e^{x^{6}}$ .

$(0.63217197, 4.38418723)$ is a local minima

Step 13