Calculus Examples

$f (x) = 10 \cdot (\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}})$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

Since $10$ is constant with respect to $x$ , the derivative of $10 (\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}})$ with respect to $x$ is $10 \frac{d}{d x} [\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}}]$ .

$10 \frac{d}{d x} [\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}}]$

Step 1.1.2

By the Sum Rule, the derivative of $\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{{(x + 3)}^{2}}] + \frac{d}{d x} [- \frac{1}{x^{2}}]$ .

$10 (\frac{d}{d x} [\frac{3}{{(x + 3)}^{2}}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.1.3

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

$10 (3 \frac{d}{d x} [\frac{1}{{(x + 3)}^{2}}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.1.4

Apply basic rules of exponents.

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Step 1.1.4.1

Rewrite $\frac{1}{{(x + 3)}^{2}}$ as ${({(x + 3)}^{2})}^{- 1}$ .

$10 (3 \frac{d}{d x} [{({(x + 3)}^{2})}^{- 1}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.1.4.2

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

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Step 1.1.4.2.1

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

$10 (3 \frac{d}{d x} [{(x + 3)}^{2 \cdot - 1}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.1.4.2.2

Multiply $2$ by $- 1$ .

$10 (3 \frac{d}{d x} [{(x + 3)}^{- 2}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

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

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x + 3$ .

$10 (3 (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.2.2

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

$10 (3 (- 2 u^{- 3} \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.2.3

Replace all occurrences of $u$ with $x + 3$ .

$10 (3 (- 2 {(x + 3)}^{- 3} \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 2$ by $3$ .

$10 (- 6 ({(x + 3)}^{- 3} \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.3.2

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

$10 (- 6 {(x + 3)}^{- 3} (\frac{d}{d x} [x] + \frac{d}{d x} [3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.3.3

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

$10 (- 6 {(x + 3)}^{- 3} (1 + \frac{d}{d x} [3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$10 (- 6 {(x + 3)}^{- 3} (1 + 0) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Add $1$ and $0$ .

$10 (- 6 {(x + 3)}^{- 3} \cdot 1 + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.3.5.2

Multiply $- 6$ by $1$ .

$10 (- 6 {(x + 3)}^{- 3} + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 1.4

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 $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 1.5

Differentiate.

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Step 1.5.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 1.5.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 1.5.2.1

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

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [x^{2 \cdot - 1}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 1.5.2.2

Multiply $2$ by $- 1$ .

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [x^{- 2}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 1.5.3

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

$10 (- 6 {(x + 3)}^{- 3} - (- 2 x^{- 3}) + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 1.5.4

Multiply $- 2$ by $- 1$ .

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3} + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 1.5.5

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

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0)$

Step 1.5.6

Simplify the expression.

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Step 1.5.6.1

Multiply $\frac{1}{x^{2}}$ by $0$ .

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3} + 0)$

Step 1.5.6.2

Add $- 6 {(x + 3)}^{- 3} + 2 x^{- 3}$ and $0$ .

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3})$

Step 1.6

Simplify.

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Step 1.6.1

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

$10 (- 6 \frac{1}{{(x + 3)}^{3}} + 2 x^{- 3})$

Step 1.6.2

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

$10 (- 6 \frac{1}{{(x + 3)}^{3}} + 2 \frac{1}{x^{3}})$

Step 1.6.3

Apply the distributive property.

$10 (- 6 \frac{1}{{(x + 3)}^{3}}) + 10 (2 \frac{1}{x^{3}})$

Step 1.6.4

Combine terms.

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Step 1.6.4.1

Combine $- 6$ and $\frac{1}{{(x + 3)}^{3}}$ .

$10 \frac{- 6}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 1.6.4.2

Move the negative in front of the fraction.

$10 (- \frac{6}{{(x + 3)}^{3}}) + 10 (2 \frac{1}{x^{3}})$

Step 1.6.4.3

Multiply $- 1$ by $10$ .

$- 10 \frac{6}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 1.6.4.4

Combine $- 10$ and $\frac{6}{{(x + 3)}^{3}}$ .

$\frac{- 10 \cdot 6}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 1.6.4.5

Multiply $- 10$ by $6$ .

$\frac{- 60}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 1.6.4.6

Move the negative in front of the fraction.

$- \frac{60}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 1.6.4.7

Combine $2$ and $\frac{1}{x^{3}}$ .

$- \frac{60}{{(x + 3)}^{3}} + 10 \frac{2}{x^{3}}$

Step 1.6.4.8

Combine $10$ and $\frac{2}{x^{3}}$ .

$- \frac{60}{{(x + 3)}^{3}} + \frac{10 \cdot 2}{x^{3}}$

Step 1.6.4.9

Multiply $10$ by $2$ .

$- \frac{60}{{(x + 3)}^{3}} + \frac{20}{x^{3}}$

Step 1.6.5

Reorder terms.

$\frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}}$ with respect to $x$ is $\frac{d}{d x} [\frac{20}{x^{3}}] + \frac{d}{d x} [- \frac{60}{{(x + 3)}^{3}}]$ .

$f'' (x) = \frac{d}{d x} (\frac{20}{x^{3}}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{20}{x^{3}}]$ .

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Step 2.2.1

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

$f'' (x) = 20 \frac{d}{d x} (\frac{1}{x^{3}}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.2

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

$f'' (x) = 20 \frac{d}{d x} ({(x^{3})}^{- 1}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.3

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 $g (x) = x^{3}$ .

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Step 2.2.3.1

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

$f'' (x) = 20 (\frac{d}{d u (1)} ({u_{1}}^{- 1}) \frac{d}{d x} (x^{3})) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.3.2

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

$f'' (x) = 20 (- {u_{1}}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.3.3

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

$f'' (x) = 20 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = 20 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 2.2.5.1

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

$f'' (x) = 20 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = 20 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.6

Multiply $3$ by $- 1$ .

$f'' (x) = 20 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 2.2.7.1

Move $x^{2}$ .

$f'' (x) = 20 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.7.2

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

$f'' (x) = 20 (- 3 x^{2 - 6}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.7.3

Subtract $6$ from $2$ .

$f'' (x) = 20 (- 3 x^{- 4}) + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.2.8

Multiply $- 3$ by $20$ .

$f'' (x) = - 60 x^{- 4} + \frac{d}{d x} (- \frac{60}{{(x + 3)}^{3}})$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{60}{{(x + 3)}^{3}}]$ .

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Step 2.3.1

Since $- 60$ is constant with respect to $x$ , the derivative of $- \frac{60}{{(x + 3)}^{3}}$ with respect to $x$ is $- 60 \frac{d}{d x} [\frac{1}{{(x + 3)}^{3}}]$ .

$f'' (x) = - 60 x^{- 4} - 60 \frac{d}{d x} \frac{1}{{(x + 3)}^{3}}$

Step 2.3.2

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

$f'' (x) = - 60 x^{- 4} - 60 \frac{d}{d x} {({(x + 3)}^{3})}^{- 1}$

Step 2.3.3

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 $g (x) = {(x + 3)}^{3}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as ${(x + 3)}^{3}$ .

$f'' (x) = - 60 x^{- 4} - 60 (\frac{d}{d u (2)} ({u_{2}}^{- 1}) \frac{d}{d x} ({(x + 3)}^{3}))$

Step 2.3.3.2

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

$f'' (x) = - 60 x^{- 4} - 60 (- {u_{2}}^{- 2} \frac{d}{d x} {(x + 3)}^{3})$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with ${(x + 3)}^{3}$ .

$f'' (x) = - 60 x^{- 4} - 60 (- {({(x + 3)}^{3})}^{- 2} \frac{d}{d x} {(x + 3)}^{3})$

Step 2.3.4

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

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Step 2.3.4.1

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

$f'' (x) = - 60 x^{- 4} - 60 (- {({(x + 3)}^{3})}^{- 2} (\frac{d}{d u (3)} ({u_{3}}^{3}) \frac{d}{d x} (x + 3)))$

Step 2.3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 3$ .

$f'' (x) = - 60 x^{- 4} - 60 (- {({(x + 3)}^{3})}^{- 2} (3 {u_{3}}^{2} \frac{d}{d x} (x + 3)))$

Step 2.3.4.3

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

$f'' (x) = - 60 x^{- 4} - 60 (- {({(x + 3)}^{3})}^{- 2} (3 {(x + 3)}^{2} \frac{d}{d x} (x + 3)))$

Step 2.3.5

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

$f'' (x) = - 60 x^{- 4} - 60 (- {({(x + 3)}^{3})}^{- 2} (3 {(x + 3)}^{2} (\frac{d}{d x} (x) + \frac{d}{d x} (3))))$

Step 2.3.6

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

$f'' (x) = - 60 x^{- 4} - 60 (- {({(x + 3)}^{3})}^{- 2} (3 {(x + 3)}^{2} (1 + \frac{d}{d x} (3))))$

Step 2.3.7

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = - 60 x^{- 4} - 60 (- {({(x + 3)}^{3})}^{- 2} (3 {(x + 3)}^{2} (1 + 0)))$

Step 2.3.8

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

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Step 2.3.8.1

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

$f'' (x) = - 60 x^{- 4} - 60 (- {(x + 3)}^{3 \cdot - 2} (3 {(x + 3)}^{2} (1 + 0)))$

Step 2.3.8.2

Multiply $3$ by $- 2$ .

$f'' (x) = - 60 x^{- 4} - 60 (- {(x + 3)}^{- 6} (3 {(x + 3)}^{2} (1 + 0)))$

Step 2.3.9

Add $1$ and $0$ .

$f'' (x) = - 60 x^{- 4} - 60 (- {(x + 3)}^{- 6} (3 {(x + 3)}^{2} \cdot 1))$

Step 2.3.10

Multiply $3$ by $1$ .

$f'' (x) = - 60 x^{- 4} - 60 (- {(x + 3)}^{- 6} (3 {(x + 3)}^{2}))$

Step 2.3.11

Multiply $3$ by $- 1$ .

$f'' (x) = - 60 x^{- 4} - 60 (- 3 {(x + 3)}^{- 6} {(x + 3)}^{2})$

Step 2.3.12

Multiply ${(x + 3)}^{- 6}$ by ${(x + 3)}^{2}$ by adding the exponents.

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Step 2.3.12.1

Move ${(x + 3)}^{2}$ .

$f'' (x) = - 60 x^{- 4} - 60 (- 3 ({(x + 3)}^{2} {(x + 3)}^{- 6}))$

Step 2.3.12.2

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

$f'' (x) = - 60 x^{- 4} - 60 (- 3 {(x + 3)}^{2 - 6})$

Step 2.3.12.3

Subtract $6$ from $2$ .

$f'' (x) = - 60 x^{- 4} - 60 (- 3 {(x + 3)}^{- 4})$

Step 2.3.13

Multiply $- 3$ by $- 60$ .

$f'' (x) = - 60 x^{- 4} + 180 {(x + 3)}^{- 4}$

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = - 60 \frac{1}{x^{4}} + 180 {(x + 3)}^{- 4}$

Step 2.4.2

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

$f'' (x) = - 60 \frac{1}{x^{4}} + 180 (\frac{1}{{(x + 3)}^{4}})$

Step 2.4.3

Combine terms.

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Step 2.4.3.1

Combine $- 60$ and $\frac{1}{x^{4}}$ .

$f'' (x) = \frac{- 60}{x^{4}} + 180 (\frac{1}{{(x + 3)}^{4}})$

Step 2.4.3.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{60}{x^{4}} + 180 (\frac{1}{{(x + 3)}^{4}})$

Step 2.4.3.3

Combine $180$ and $\frac{1}{{(x + 3)}^{4}}$ .

$f'' (x) = - \frac{60}{x^{4}} + \frac{180}{{(x + 3)}^{4}}$

Step 2.4.4

Reorder terms.

$f'' (x) = \frac{180}{{(x + 3)}^{4}} - \frac{60}{x^{4}}$

Step 3

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

$\frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}} = 0$

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

Differentiate.

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Step 4.1.1.1

Since $10$ is constant with respect to $x$ , the derivative of $10 (\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}})$ with respect to $x$ is $10 \frac{d}{d x} [\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}}]$ .

$10 \frac{d}{d x} [\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}}]$

Step 4.1.1.2

By the Sum Rule, the derivative of $\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{{(x + 3)}^{2}}] + \frac{d}{d x} [- \frac{1}{x^{2}}]$ .

$10 (\frac{d}{d x} [\frac{3}{{(x + 3)}^{2}}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.1.3

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

$10 (3 \frac{d}{d x} [\frac{1}{{(x + 3)}^{2}}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.1.4

Apply basic rules of exponents.

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Step 4.1.1.4.1

Rewrite $\frac{1}{{(x + 3)}^{2}}$ as ${({(x + 3)}^{2})}^{- 1}$ .

$10 (3 \frac{d}{d x} [{({(x + 3)}^{2})}^{- 1}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.1.4.2

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

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Step 4.1.1.4.2.1

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

$10 (3 \frac{d}{d x} [{(x + 3)}^{2 \cdot - 1}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.1.4.2.2

Multiply $2$ by $- 1$ .

$10 (3 \frac{d}{d x} [{(x + 3)}^{- 2}] + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.2

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

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Step 4.1.2.1

To apply the Chain Rule, set $u$ as $x + 3$ .

$10 (3 (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.2.2

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

$10 (3 (- 2 u^{- 3} \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.2.3

Replace all occurrences of $u$ with $x + 3$ .

$10 (3 (- 2 {(x + 3)}^{- 3} \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

Multiply $- 2$ by $3$ .

$10 (- 6 ({(x + 3)}^{- 3} \frac{d}{d x} [x + 3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.3.2

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

$10 (- 6 {(x + 3)}^{- 3} (\frac{d}{d x} [x] + \frac{d}{d x} [3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.3.3

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

$10 (- 6 {(x + 3)}^{- 3} (1 + \frac{d}{d x} [3]) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.3.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$10 (- 6 {(x + 3)}^{- 3} (1 + 0) + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.3.5

Simplify the expression.

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Step 4.1.3.5.1

Add $1$ and $0$ .

$10 (- 6 {(x + 3)}^{- 3} \cdot 1 + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.3.5.2

Multiply $- 6$ by $1$ .

$10 (- 6 {(x + 3)}^{- 3} + \frac{d}{d x} [- \frac{1}{x^{2}}])$

Step 4.1.4

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 $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 4.1.5

Differentiate.

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Step 4.1.5.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 4.1.5.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.1.5.2.1

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

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [x^{2 \cdot - 1}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 4.1.5.2.2

Multiply $2$ by $- 1$ .

$10 (- 6 {(x + 3)}^{- 3} - \frac{d}{d x} [x^{- 2}] + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 4.1.5.3

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

$10 (- 6 {(x + 3)}^{- 3} - (- 2 x^{- 3}) + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 4.1.5.4

Multiply $- 2$ by $- 1$ .

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3} + \frac{1}{x^{2}} \frac{d}{d x} [- 1])$

Step 4.1.5.5

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

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3} + \frac{1}{x^{2}} \cdot 0)$

Step 4.1.5.6

Simplify the expression.

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Step 4.1.5.6.1

Multiply $\frac{1}{x^{2}}$ by $0$ .

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3} + 0)$

Step 4.1.5.6.2

Add $- 6 {(x + 3)}^{- 3} + 2 x^{- 3}$ and $0$ .

$10 (- 6 {(x + 3)}^{- 3} + 2 x^{- 3})$

Step 4.1.6

Simplify.

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Step 4.1.6.1

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

$10 (- 6 \frac{1}{{(x + 3)}^{3}} + 2 x^{- 3})$

Step 4.1.6.2

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

$10 (- 6 \frac{1}{{(x + 3)}^{3}} + 2 \frac{1}{x^{3}})$

Step 4.1.6.3

Apply the distributive property.

$10 (- 6 \frac{1}{{(x + 3)}^{3}}) + 10 (2 \frac{1}{x^{3}})$

Step 4.1.6.4

Combine terms.

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Step 4.1.6.4.1

Combine $- 6$ and $\frac{1}{{(x + 3)}^{3}}$ .

$10 \frac{- 6}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 4.1.6.4.2

Move the negative in front of the fraction.

$10 (- \frac{6}{{(x + 3)}^{3}}) + 10 (2 \frac{1}{x^{3}})$

Step 4.1.6.4.3

Multiply $- 1$ by $10$ .

$- 10 \frac{6}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 4.1.6.4.4

Combine $- 10$ and $\frac{6}{{(x + 3)}^{3}}$ .

$\frac{- 10 \cdot 6}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 4.1.6.4.5

Multiply $- 10$ by $6$ .

$\frac{- 60}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 4.1.6.4.6

Move the negative in front of the fraction.

$- \frac{60}{{(x + 3)}^{3}} + 10 (2 \frac{1}{x^{3}})$

Step 4.1.6.4.7

Combine $2$ and $\frac{1}{x^{3}}$ .

$- \frac{60}{{(x + 3)}^{3}} + 10 \frac{2}{x^{3}}$

Step 4.1.6.4.8

Combine $10$ and $\frac{2}{x^{3}}$ .

$- \frac{60}{{(x + 3)}^{3}} + \frac{10 \cdot 2}{x^{3}}$

Step 4.1.6.4.9

Multiply $10$ by $2$ .

$- \frac{60}{{(x + 3)}^{3}} + \frac{20}{x^{3}}$

Step 4.1.6.5

Reorder terms.

$f' (x) = \frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}}$ .

$\frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{20}{x^{3}} - \frac{60}{{(x + 3)}^{3}} = 0$

Step 5.2

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

$x \approx 6.78350009$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{20}{x^{3}}$ equal to $0$ to find where the expression is undefined.

$x^{3} = 0$

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.

$x = \sqrt[3]{0}$

Step 6.2.2

Simplify $\sqrt[3]{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 6.2.2.2

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

$x = 0$

Step 6.3

Set the denominator in $\frac{60}{{(x + 3)}^{3}}$ equal to $0$ to find where the expression is undefined.

${(x + 3)}^{3} = 0$

Step 6.4

Solve for $x$ .

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Step 6.4.1

Set the $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 6.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6.5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 3, x = 0$

Step 7

Critical points to evaluate.

$x = 6.78350009$

Step 8

Evaluate the second derivative at $x = 6.78350009$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{180}{{((6.78350009) + 3)}^{4}} - \frac{60}{{(6.78350009)}^{4}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Simplify the denominator.

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Step 9.1.1.1

Add $6.78350009$ and $3$ .

$\frac{180}{{9.78350009}^{4}} - \frac{60}{{(6.78350009)}^{4}}$

Step 9.1.1.2

Raise $9.78350009$ to the power of $4$ .

$\frac{180}{9161.72000483} - \frac{60}{{(6.78350009)}^{4}}$

Step 9.1.2

Divide $180$ by $9161.72000483$ .

$0.01964696 - \frac{60}{{(6.78350009)}^{4}}$

Step 9.1.3

Raise $6.78350009$ to the power of $4$ .

$0.01964696 - \frac{60}{2117.46062276}$

Step 9.1.4

Divide $60$ by $2117.46062276$ .

$0.01964696 - 1 \cdot 0.02833582$

Step 9.1.5

Multiply $- 1$ by $0.02833582$ .

$0.01964696 - 0.02833582$

Step 9.2

Subtract $0.02833582$ from $0.01964696$ .

$- 0.00868886$

Step 10

$x = 6.78350009$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 6.78350009$ is a local maximum

Step 11

Find the y-value when $x = 6.78350009$ .

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Step 11.1

Replace the variable $x$ with $6.78350009$ in the expression.

$f (6.78350009) = 10 \cdot (\frac{3}{{((6.78350009) + 3)}^{2}} - \frac{1}{{(6.78350009)}^{2}})$

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

Simplify the denominator.

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Step 11.2.1.1.1

Add $6.78350009$ and $3$ .

$f (6.78350009) = 10 \cdot (\frac{3}{{9.78350009}^{2}} - \frac{1}{{(6.78350009)}^{2}})$

Step 11.2.1.1.2

Raise $9.78350009$ to the power of $2$ .

$f (6.78350009) = 10 \cdot (\frac{3}{95.71687419} - \frac{1}{{(6.78350009)}^{2}})$

Step 11.2.1.2

Divide $3$ by $95.71687419$ .

$f (6.78350009) = 10 \cdot (0.03134243 - \frac{1}{{(6.78350009)}^{2}})$

Step 11.2.1.3

Raise $6.78350009$ to the power of $2$ .

$f (6.78350009) = 10 \cdot (0.03134243 - \frac{1}{46.01587359})$

Step 11.2.1.4

Divide $1$ by $46.01587359$ .

$f (6.78350009) = 10 \cdot (0.03134243 - 1 \cdot 0.02173163)$

Step 11.2.1.5

Multiply $- 1$ by $0.02173163$ .

$f (6.78350009) = 10 \cdot (0.03134243 - 0.02173163)$

Step 11.2.2

Simplify the expression.

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Step 11.2.2.1

Subtract $0.02173163$ from $0.03134243$ .

$f (6.78350009) = 10 \cdot 0.0096108$

Step 11.2.2.2

Multiply $10$ by $0.0096108$ .

$f (6.78350009) = 0.09610804$

Step 11.2.3

The final answer is $0.09610804$ .

$y = 0.09610804$

Step 12

These are the local extrema for $f (x) = 10 \cdot (\frac{3}{{(x + 3)}^{2}} - \frac{1}{x^{2}})$ .

$(6.78350009, 0.09610804)$ is a local maxima

Step 13