Algebra Examples

$f (x) = \frac{20}{1 + 9 e^{- 3 x}}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $20$ is constant with respect to $x$ , the derivative of $\frac{20}{1 + 9 e^{- 3 x}}$ with respect to $x$ is $20 \frac{d}{d x} [\frac{1}{1 + 9 e^{- 3 x}}]$ .

$20 \frac{d}{d x} [\frac{1}{1 + 9 e^{- 3 x}}]$

Step 1.1.2

Rewrite $\frac{1}{1 + 9 e^{- 3 x}}$ as ${(1 + 9 e^{- 3 x})}^{- 1}$ .

$20 \frac{d}{d x} [{(1 + 9 e^{- 3 x})}^{- 1}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 1 + 9 e^{- 3 x}$ .

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Step 1.2.1

To apply the Chain Rule, set $u_{1}$ as $1 + 9 e^{- 3 x}$ .

$20 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [1 + 9 e^{- 3 x}])$

Step 1.2.2

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

$20 (- {u_{1}}^{- 2} \frac{d}{d x} [1 + 9 e^{- 3 x}])$

Step 1.2.3

Replace all occurrences of $u_{1}$ with $1 + 9 e^{- 3 x}$ .

$20 (- {(1 + 9 e^{- 3 x})}^{- 2} \frac{d}{d x} [1 + 9 e^{- 3 x}])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $20$ .

$- 20 ({(1 + 9 e^{- 3 x})}^{- 2} \frac{d}{d x} [1 + 9 e^{- 3 x}])$

Step 1.3.2

By the Sum Rule, the derivative of $1 + 9 e^{- 3 x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [9 e^{- 3 x}]$ .

$- 20 {(1 + 9 e^{- 3 x})}^{- 2} (\frac{d}{d x} [1] + \frac{d}{d x} [9 e^{- 3 x}])$

Step 1.3.3

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

$- 20 {(1 + 9 e^{- 3 x})}^{- 2} (0 + \frac{d}{d x} [9 e^{- 3 x}])$

Step 1.3.4

Add $0$ and $\frac{d}{d x} [9 e^{- 3 x}]$ .

$- 20 {(1 + 9 e^{- 3 x})}^{- 2} \frac{d}{d x} [9 e^{- 3 x}]$

Step 1.3.5

Since $9$ is constant with respect to $x$ , the derivative of $9 e^{- 3 x}$ with respect to $x$ is $9 \frac{d}{d x} [e^{- 3 x}]$ .

$- 20 {(1 + 9 e^{- 3 x})}^{- 2} (9 \frac{d}{d x} [e^{- 3 x}])$

Step 1.3.6

Multiply $9$ by $- 20$ .

$- 180 {(1 + 9 e^{- 3 x})}^{- 2} \frac{d}{d x} [e^{- 3 x}]$

Step 1.4

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 $g (x) = - 3 x$ .

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Step 1.4.1

To apply the Chain Rule, set $u_{2}$ as $- 3 x$ .

$- 180 {(1 + 9 e^{- 3 x})}^{- 2} (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [- 3 x])$

Step 1.4.2

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

$- 180 {(1 + 9 e^{- 3 x})}^{- 2} (e^{u_{2}} \frac{d}{d x} [- 3 x])$

Step 1.4.3

Replace all occurrences of $u_{2}$ with $- 3 x$ .

$- 180 {(1 + 9 e^{- 3 x})}^{- 2} (e^{- 3 x} \frac{d}{d x} [- 3 x])$

Step 1.5

Differentiate.

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Step 1.5.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 180 {(1 + 9 e^{- 3 x})}^{- 2} (e^{- 3 x} (- 3 \frac{d}{d x} [x]))$

Step 1.5.2

Multiply $- 3$ by $- 180$ .

$540 {(1 + 9 e^{- 3 x})}^{- 2} (e^{- 3 x} (\frac{d}{d x} [x]))$

Step 1.5.3

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

$540 {(1 + 9 e^{- 3 x})}^{- 2} (e^{- 3 x} \cdot 1)$

Step 1.5.4

Multiply $e^{- 3 x}$ by $1$ .

$540 {(1 + 9 e^{- 3 x})}^{- 2} e^{- 3 x}$

Step 1.6

Simplify.

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Step 1.6.1

Reorder the factors of $540 {(1 + 9 e^{- 3 x})}^{- 2} e^{- 3 x}$ .

$540 e^{- 3 x} {(1 + 9 e^{- 3 x})}^{- 2}$

Step 1.6.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$540 e^{- 3 x} \frac{1}{{(1 + 9 e^{- 3 x})}^{2}}$

Step 1.6.3

Multiply $540 e^{- 3 x} \frac{1}{{(1 + 9 e^{- 3 x})}^{2}}$ .

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Step 1.6.3.1

Combine $\frac{1}{{(1 + 9 e^{- 3 x})}^{2}}$ and $540$ .

$\frac{540}{{(1 + 9 e^{- 3 x})}^{2}} e^{- 3 x}$

Step 1.6.3.2

Combine $\frac{540}{{(1 + 9 e^{- 3 x})}^{2}}$ and $e^{- 3 x}$ .

$\frac{540 e^{- 3 x}}{{(1 + 9 e^{- 3 x})}^{2}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $540$ is constant with respect to $x$ , the derivative of $\frac{540 e^{- 3 x}}{{(1 + 9 e^{- 3 x})}^{2}}$ with respect to $x$ is $540 \frac{d}{d x} [\frac{e^{- 3 x}}{{(1 + 9 e^{- 3 x})}^{2}}]$ .

$f'' (x) = 540 \frac{d}{d x} (\frac{e^{- 3 x}}{{(1 + 9 e^{- 3 x})}^{2}})$

Step 2.2

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 $f (x) = e^{- 3 x}$ and $g (x) = {(1 + 9 e^{- 3 x})}^{2}$ .

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} \frac{d}{d x} (e^{- 3 x}) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{({(1 + 9 e^{- 3 x})}^{2})}^{2}})$

Step 2.3

Multiply the exponents in ${({(1 + 9 e^{- 3 x})}^{2})}^{2}$ .

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Step 2.3.1

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

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} \frac{d}{d x} (e^{- 3 x}) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{2 \cdot 2}})$

Step 2.3.2

Multiply $2$ by $2$ .

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} \frac{d}{d x} (e^{- 3 x}) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

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 $g (x) = - 3 x$ .

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Step 2.4.1

To apply the Chain Rule, set $u_{1}$ as $- 3 x$ .

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} (\frac{d}{d u (1)} (e^{u_{1}}) \frac{d}{d x} (- 3 x)) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

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 $a$ = $e$ .

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} (e^{u_{1}} \frac{d}{d x} (- 3 x)) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.4.3

Replace all occurrences of $u_{1}$ with $- 3 x$ .

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} (e^{- 3 x} \frac{d}{d x} (- 3 x)) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.5

Differentiate.

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Step 2.5.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} (- 3 \frac{d}{d x} x) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.5.2

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

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} (- 3 \cdot 1) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.5.3

Simplify the expression.

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Step 2.5.3.1

Multiply $- 3$ by $1$ .

$f'' (x) = 540 (\frac{{(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} \cdot - 3 - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.5.3.2

Move $- 3$ to the left of ${(1 + 9 e^{- 3 x})}^{2} e^{- 3 x}$ .

$f'' (x) = 540 (\frac{- 3 \cdot ({(1 + 9 e^{- 3 x})}^{2} e^{- 3 x}) - e^{- 3 x} \frac{d}{d x} {(1 + 9 e^{- 3 x})}^{2}}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 1 + 9 e^{- 3 x}$ .

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Step 2.6.1

To apply the Chain Rule, set $u_{2}$ as $1 + 9 e^{- 3 x}$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - e^{- 3 x} (\frac{d}{d u (2)} ({u_{2}}^{2}) \frac{d}{d x} (1 + 9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.6.2

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

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - e^{- 3 x} (2 u_{2} \frac{d}{d x} (1 + 9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.6.3

Replace all occurrences of $u_{2}$ with $1 + 9 e^{- 3 x}$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - e^{- 3 x} (2 (1 + 9 e^{- 3 x}) \frac{d}{d x} (1 + 9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.7

Differentiate.

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Step 2.7.1

Multiply $2$ by $- 1$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 2 e^{- 3 x} ((1 + 9 e^{- 3 x}) \frac{d}{d x} (1 + 9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.7.2

By the Sum Rule, the derivative of $1 + 9 e^{- 3 x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [9 e^{- 3 x}]$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 2 e^{- 3 x} ((1 + 9 e^{- 3 x}) (\frac{d}{d x} (1) + \frac{d}{d x} (9 e^{- 3 x})))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.7.3

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

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 2 e^{- 3 x} ((1 + 9 e^{- 3 x}) (0 + \frac{d}{d x} (9 e^{- 3 x})))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.7.4

Add $0$ and $\frac{d}{d x} [9 e^{- 3 x}]$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 2 e^{- 3 x} ((1 + 9 e^{- 3 x}) \frac{d}{d x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.7.5

Since $9$ is constant with respect to $x$ , the derivative of $9 e^{- 3 x}$ with respect to $x$ is $9 \frac{d}{d x} [e^{- 3 x}]$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 2 e^{- 3 x} ((1 + 9 e^{- 3 x}) (9 \frac{d}{d x} (e^{- 3 x})))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.7.6

Simplify the expression.

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Step 2.7.6.1

Move $9$ to the left of $1 + 9 e^{- 3 x}$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 2 e^{- 3 x} (9 \cdot (1 + 9 e^{- 3 x}) \frac{d}{d x} (e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.7.6.2

Multiply $9$ by $- 2$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 18 e^{- 3 x} ((1 + 9 e^{- 3 x}) \frac{d}{d x} (e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.8

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 $g (x) = - 3 x$ .

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Step 2.8.1

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

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 18 e^{- 3 x} ((1 + 9 e^{- 3 x}) (\frac{d}{d u (3)} (e^{u_{3}}) \frac{d}{d x} (- 3 x)))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.8.2

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

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 18 e^{- 3 x} ((1 + 9 e^{- 3 x}) (e^{u_{3}} \frac{d}{d x} (- 3 x)))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.8.3

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

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 18 e^{- 3 x} ((1 + 9 e^{- 3 x}) (e^{- 3 x} \frac{d}{d x} (- 3 x)))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.9

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

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 18 e^{- 3 x - 3 x} ((1 + 9 e^{- 3 x}) \frac{d}{d x} (- 3 x))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.10

Subtract $3 x$ from $- 3 x$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 18 e^{- 6 x} ((1 + 9 e^{- 3 x}) \frac{d}{d x} (- 3 x))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.11

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} - 18 e^{- 6 x} ((1 + 9 e^{- 3 x}) (- 3 \frac{d}{d x} x))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.12

Multiply $- 3$ by $- 18$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} + 54 e^{- 6 x} ((1 + 9 e^{- 3 x}) (\frac{d}{d x} (x)))}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.13

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

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} + 54 e^{- 6 x} ((1 + 9 e^{- 3 x}) \cdot 1)}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.14

Combine fractions.

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Step 2.14.1

Multiply $1 + 9 e^{- 3 x}$ by $1$ .

$f'' (x) = 540 (\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} + 54 e^{- 6 x} (1 + 9 e^{- 3 x})}{{(1 + 9 e^{- 3 x})}^{4}})$

Step 2.14.2

Combine $540$ and $\frac{- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} + 54 e^{- 6 x} (1 + 9 e^{- 3 x})}{{(1 + 9 e^{- 3 x})}^{4}}$ .

$f'' (x) = \frac{540 (- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} + 54 e^{- 6 x} (1 + 9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15

Simplify.

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Step 2.15.1

Apply the distributive property.

$f'' (x) = \frac{540 (- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x} + 54 e^{- 6 x} \cdot 1 + 54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.2

Apply the distributive property.

$f'' (x) = \frac{540 (- 3 {(1 + 9 e^{- 3 x})}^{2} e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3

Simplify the numerator.

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Step 2.15.3.1

Simplify each term.

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Step 2.15.3.1.1

Rewrite ${(1 + 9 e^{- 3 x})}^{2}$ as $(1 + 9 e^{- 3 x}) (1 + 9 e^{- 3 x})$ .

$f'' (x) = \frac{540 (- 3 ((1 + 9 e^{- 3 x}) (1 + 9 e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.2

Expand $(1 + 9 e^{- 3 x}) (1 + 9 e^{- 3 x})$ using the FOIL Method.

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Step 2.15.3.1.2.1

Apply the distributive property.

$f'' (x) = \frac{540 (- 3 (1 (1 + 9 e^{- 3 x}) + 9 e^{- 3 x} (1 + 9 e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.2.2

Apply the distributive property.

$f'' (x) = \frac{540 (- 3 (1 \cdot 1 + 1 (9 e^{- 3 x}) + 9 e^{- 3 x} (1 + 9 e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.2.3

Apply the distributive property.

$f'' (x) = \frac{540 (- 3 (1 \cdot 1 + 1 (9 e^{- 3 x}) + 9 e^{- 3 x} \cdot 1 + 9 e^{- 3 x} (9 e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3

Simplify and combine like terms.

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Step 2.15.3.1.3.1

Simplify each term.

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Step 2.15.3.1.3.1.1

Multiply $1$ by $1$ .

$f'' (x) = \frac{540 (- 3 (1 + 1 (9 e^{- 3 x}) + 9 e^{- 3 x} \cdot 1 + 9 e^{- 3 x} (9 e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.1.2

Multiply $9 e^{- 3 x}$ by $1$ .

$f'' (x) = \frac{540 (- 3 (1 + 9 e^{- 3 x} + 9 e^{- 3 x} \cdot 1 + 9 e^{- 3 x} (9 e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.1.3

Multiply $9$ by $1$ .

$f'' (x) = \frac{540 (- 3 (1 + 9 e^{- 3 x} + 9 e^{- 3 x} + 9 e^{- 3 x} (9 e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{540 (- 3 (1 + 9 e^{- 3 x} + 9 e^{- 3 x} + 9 \cdot (9 e^{- 3 x} e^{- 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.1.5

Multiply $e^{- 3 x}$ by $e^{- 3 x}$ by adding the exponents.

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Step 2.15.3.1.3.1.5.1

Move $e^{- 3 x}$ .

$f'' (x) = \frac{540 (- 3 (1 + 9 e^{- 3 x} + 9 e^{- 3 x} + 9 \cdot (9 (e^{- 3 x} e^{- 3 x}))) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.1.5.2

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

$f'' (x) = \frac{540 (- 3 (1 + 9 e^{- 3 x} + 9 e^{- 3 x} + 9 \cdot (9 e^{- 3 x - 3 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.1.5.3

Subtract $3 x$ from $- 3 x$ .

$f'' (x) = \frac{540 (- 3 (1 + 9 e^{- 3 x} + 9 e^{- 3 x} + 9 \cdot (9 e^{- 6 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.1.6

Multiply $9$ by $9$ .

$f'' (x) = \frac{540 (- 3 (1 + 9 e^{- 3 x} + 9 e^{- 3 x} + 81 e^{- 6 x}) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.3.2

Add $9 e^{- 3 x}$ and $9 e^{- 3 x}$ .

$f'' (x) = \frac{540 (- 3 (1 + 18 e^{- 3 x} + 81 e^{- 6 x}) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.4

Apply the distributive property.

$f'' (x) = \frac{540 ((- 3 \cdot 1 - 3 (18 e^{- 3 x}) - 3 (81 e^{- 6 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.5

Simplify.

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Step 2.15.3.1.5.1

Multiply $- 3$ by $1$ .

$f'' (x) = \frac{540 ((- 3 - 3 (18 e^{- 3 x}) - 3 (81 e^{- 6 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.5.2

Multiply $18$ by $- 3$ .

$f'' (x) = \frac{540 ((- 3 - 54 e^{- 3 x} - 3 (81 e^{- 6 x})) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.5.3

Multiply $81$ by $- 3$ .

$f'' (x) = \frac{540 ((- 3 - 54 e^{- 3 x} - 243 e^{- 6 x}) e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.6

Apply the distributive property.

$f'' (x) = \frac{540 (- 3 e^{- 3 x} - 54 e^{- 3 x} e^{- 3 x} - 243 e^{- 6 x} e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.7

Simplify.

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Step 2.15.3.1.7.1

Multiply $e^{- 3 x}$ by $e^{- 3 x}$ by adding the exponents.

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Step 2.15.3.1.7.1.1

Move $e^{- 3 x}$ .

$f'' (x) = \frac{540 (- 3 e^{- 3 x} - 54 (e^{- 3 x} e^{- 3 x}) - 243 e^{- 6 x} e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.7.1.2

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

$f'' (x) = \frac{540 (- 3 e^{- 3 x} - 54 e^{- 3 x - 3 x} - 243 e^{- 6 x} e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.7.1.3

Subtract $3 x$ from $- 3 x$ .

$f'' (x) = \frac{540 (- 3 e^{- 3 x} - 54 e^{- 6 x} - 243 e^{- 6 x} e^{- 3 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.7.2

Multiply $e^{- 6 x}$ by $e^{- 3 x}$ by adding the exponents.

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Step 2.15.3.1.7.2.1

Move $e^{- 3 x}$ .

$f'' (x) = \frac{540 (- 3 e^{- 3 x} - 54 e^{- 6 x} - 243 (e^{- 3 x} e^{- 6 x})) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.7.2.2

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

$f'' (x) = \frac{540 (- 3 e^{- 3 x} - 54 e^{- 6 x} - 243 e^{- 3 x - 6 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.7.2.3

Subtract $6 x$ from $- 3 x$ .

$f'' (x) = \frac{540 (- 3 e^{- 3 x} - 54 e^{- 6 x} - 243 e^{- 9 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.8

Apply the distributive property.

$f'' (x) = \frac{540 (- 3 e^{- 3 x}) + 540 (- 54 e^{- 6 x}) + 540 (- 243 e^{- 9 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.9

Simplify.

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Step 2.15.3.1.9.1

Multiply $- 3$ by $540$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} + 540 (- 54 e^{- 6 x}) + 540 (- 243 e^{- 9 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.9.2

Multiply $- 54$ by $540$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} + 540 (- 243 e^{- 9 x}) + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.9.3

Multiply $- 243$ by $540$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 540 (54 e^{- 6 x} \cdot 1) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.10

Multiply $54$ by $1$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 540 (54 e^{- 6 x}) + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.11

Multiply $54$ by $540$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 29160 e^{- 6 x} + 540 (54 e^{- 6 x} (9 e^{- 3 x}))}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.12

Multiply $e^{- 6 x}$ by $e^{- 3 x}$ by adding the exponents.

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Step 2.15.3.1.12.1

Move $e^{- 3 x}$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 29160 e^{- 6 x} + 540 (54 (e^{- 3 x} e^{- 6 x}) \cdot 9)}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.12.2

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

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 29160 e^{- 6 x} + 540 (54 e^{- 3 x - 6 x} \cdot 9)}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.12.3

Subtract $6 x$ from $- 3 x$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 29160 e^{- 6 x} + 540 (54 e^{- 9 x} \cdot 9)}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.13

Multiply $9$ by $54$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 29160 e^{- 6 x} + 540 (486 e^{- 9 x})}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.1.14

Multiply $486$ by $540$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 29160 e^{- 6 x} + 262440 e^{- 9 x}}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.2

Combine the opposite terms in $- 1620 e^{- 3 x} - 29160 e^{- 6 x} - 131220 e^{- 9 x} + 29160 e^{- 6 x} + 262440 e^{- 9 x}$ .

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Step 2.15.3.2.1

Add $- 29160 e^{- 6 x}$ and $29160 e^{- 6 x}$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} + 0 - 131220 e^{- 9 x} + 262440 e^{- 9 x}}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.2.2

Add $- 1620 e^{- 3 x}$ and $0$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} - 131220 e^{- 9 x} + 262440 e^{- 9 x}}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.3.3

Add $- 131220 e^{- 9 x}$ and $262440 e^{- 9 x}$ .

$f'' (x) = \frac{- 1620 e^{- 3 x} + 131220 e^{- 9 x}}{{(1 + 9 e^{- 3 x})}^{4}}$

Step 2.15.4

Reorder terms.

$f'' (x) = \frac{131220 e^{- 9 x} - 1620 e^{- 3 x}}{{(9 e^{- 3 x} + 1)}^{4}}$

Step 2.15.5

Factor $1620$ out of $131220 e^{- 9 x} - 1620 e^{- 3 x}$ .

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Step 2.15.5.1

Factor $1620$ out of $131220 e^{- 9 x}$ .

$f'' (x) = \frac{1620 (81 e^{- 9 x}) - 1620 e^{- 3 x}}{{(9 e^{- 3 x} + 1)}^{4}}$

Step 2.15.5.2

Factor $1620$ out of $- 1620 e^{- 3 x}$ .

$f'' (x) = \frac{1620 (81 e^{- 9 x}) + 1620 (- e^{- 3 x})}{{(9 e^{- 3 x} + 1)}^{4}}$

Step 2.15.5.3

Factor $1620$ out of $1620 (81 e^{- 9 x}) + 1620 (- e^{- 3 x})$ .

$f'' (x) = \frac{1620 (81 e^{- 9 x} - e^{- 3 x})}{{(9 e^{- 3 x} + 1)}^{4}}$

Step 3

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

$\frac{540 e^{- 3 x}}{{(1 + 9 e^{- 3 x})}^{2}} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6