Calculus Examples

$f (x) = 600 (1 - \frac{7}{x + 1} + \frac{14}{{(x + 1)}^{2}})$

Step 1

Find the first derivative of the function.

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Step 1.1

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

$\frac{d}{d x} [600 (\frac{x + 1}{x + 1} - \frac{7}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.2

Combine the numerators over the common denominator.

$\frac{d}{d x} [600 (\frac{x + 1 - 7}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.3

Subtract $7$ from $1$ .

$\frac{d}{d x} [600 (\frac{x - 6}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.4

To write $\frac{x - 6}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{d}{d x} [600 (\frac{x - 6}{x + 1} \cdot \frac{x + 1}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.5

Write each expression with a common denominator of ${(x + 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

Multiply $\frac{x - 6}{x + 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{(x + 1) (x + 1)} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.5.2

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

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{1} (x + 1)} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.5.3

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

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{1} {(x + 1)}^{1}} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.5.4

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

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{1 + 1}} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.5.5

Add $1$ and $1$ .

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{2}} + \frac{14}{{(x + 1)}^{2}})]$

Step 1.6

Differentiate using the Constant Multiple Rule.

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Step 1.6.1

Combine the numerators over the common denominator.

$\frac{d}{d x} [600 \frac{(x - 6) (x + 1) + 14}{{(x + 1)}^{2}}]$

Step 1.6.2

Combine $600$ and $\frac{(x - 6) (x + 1) + 14}{{(x + 1)}^{2}}$ .

$\frac{d}{d x} [\frac{600 ((x - 6) (x + 1) + 14)}{{(x + 1)}^{2}}]$

Step 1.6.3

Since $600$ is constant with respect to $x$ , the derivative of $\frac{600 ((x - 6) (x + 1) + 14)}{{(x + 1)}^{2}}$ with respect to $x$ is $600 \frac{d}{d x} [\frac{(x - 6) (x + 1) + 14}{{(x + 1)}^{2}}]$ .

$600 \frac{d}{d x} [\frac{(x - 6) (x + 1) + 14}{{(x + 1)}^{2}}]$

Step 1.7

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 $f (x) = (x - 6) (x + 1) + 14$ and $g (x) = {(x + 1)}^{2}$ .

$600 \frac{{(x + 1)}^{2} \frac{d}{d x} [(x - 6) (x + 1) + 14] - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{({(x + 1)}^{2})}^{2}}$

Step 1.8

Differentiate using the Sum Rule.

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Step 1.8.1

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

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Step 1.8.1.1

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

$600 \frac{{(x + 1)}^{2} \frac{d}{d x} [(x - 6) (x + 1) + 14] - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{2 \cdot 2}}$

Step 1.8.1.2

Multiply $2$ by $2$ .

$600 \frac{{(x + 1)}^{2} \frac{d}{d x} [(x - 6) (x + 1) + 14] - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.8.2

By the Sum Rule, the derivative of $(x - 6) (x + 1) + 14$ with respect to $x$ is $\frac{d}{d x} [(x - 6) (x + 1)] + \frac{d}{d x} [14]$ .

$600 \frac{{(x + 1)}^{2} (\frac{d}{d x} [(x - 6) (x + 1)] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.9

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 $g (x) = x + 1$ .

$600 \frac{{(x + 1)}^{2} ((x - 6) \frac{d}{d x} [x + 1] + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10

Differentiate.

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Step 1.10.1

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

$600 \frac{{(x + 1)}^{2} ((x - 6) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.2

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

$600 \frac{{(x + 1)}^{2} ((x - 6) (1 + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.3

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

$600 \frac{{(x + 1)}^{2} ((x - 6) (1 + 0) + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.4

Simplify the expression.

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Step 1.10.4.1

Add $1$ and $0$ .

$600 \frac{{(x + 1)}^{2} ((x - 6) \cdot 1 + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.4.2

Multiply $x - 6$ by $1$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.5

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

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]) + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.6

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

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) (1 + \frac{d}{d x} [- 6]) + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) (1 + 0) + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.8

Simplify by adding terms.

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Step 1.10.8.1

Add $1$ and $0$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) \cdot 1 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.8.2

Multiply $x + 1$ by $1$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + x + 1 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.8.3

Add $x$ and $x$ .

$600 \frac{{(x + 1)}^{2} (2 x - 6 + 1 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.8.4

Add $- 6$ and $1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.9

Since $14$ is constant with respect to $x$ , the derivative of $14$ with respect to $x$ is $0$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5 + 0) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.10.10

Add $2 x - 5$ and $0$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 1.11

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 $g (x) = x + 1$ .

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Step 1.11.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 1.11.2

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

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) (2 u \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 1.11.3

Replace all occurrences of $u$ with $x + 1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) (2 (x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 1.12

Simplify with factoring out.

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Step 1.12.1

Multiply $2$ by $- 1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 1.12.2

Factor $x + 1$ out of ${(x + 1)}^{2} (2 x - 5) - 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])$ .

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Step 1.12.2.1

Factor $x + 1$ out of ${(x + 1)}^{2} (2 x - 5)$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5)) - 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 1.12.2.2

Factor $x + 1$ out of $- 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5)) + (x + 1) (- 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{{(x + 1)}^{4}}$

Step 1.12.2.3

Factor $x + 1$ out of $(x + 1) ((x + 1) (2 x - 5)) + (x + 1) (- 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{{(x + 1)}^{4}}$

Step 1.13

Cancel the common factors.

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Step 1.13.1

Factor $x + 1$ out of ${(x + 1)}^{4}$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{(x + 1) {(x + 1)}^{3}}$

Step 1.13.2

Cancel the common factor.

$600 \frac{(x + 1) ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{(x + 1) {(x + 1)}^{3}}$

Step 1.13.3

Rewrite the expression.

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1])}{{(x + 1)}^{3}}$

Step 1.14

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

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(x + 1)}^{3}}$

Step 1.15

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

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (1 + \frac{d}{d x} [1])}{{(x + 1)}^{3}}$

Step 1.16

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

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (1 + 0)}{{(x + 1)}^{3}}$

Step 1.17

Combine fractions.

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Step 1.17.1

Add $1$ and $0$ .

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) \cdot 1}{{(x + 1)}^{3}}$

Step 1.17.2

Multiply $- 2$ by $1$ .

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14)}{{(x + 1)}^{3}}$

Step 1.17.3

Combine $600$ and $\frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14)}{{(x + 1)}^{3}}$ .

$\frac{600 ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14))}{{(x + 1)}^{3}}$

Step 1.18

Simplify.

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Step 1.18.1

Apply the distributive property.

$\frac{600 ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1)) - 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.2

Apply the distributive property.

$\frac{600 ((x + 1) (2 x - 5)) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3

Simplify the numerator.

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Step 1.18.3.1

Simplify each term.

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Step 1.18.3.1.1

Expand $(x + 1) (2 x - 5)$ using the FOIL Method.

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Step 1.18.3.1.1.1

Apply the distributive property.

$\frac{600 (x (2 x - 5) + 1 (2 x - 5)) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.1.2

Apply the distributive property.

$\frac{600 (x (2 x) + x \cdot - 5 + 1 (2 x - 5)) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.1.3

Apply the distributive property.

$\frac{600 (x (2 x) + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.2

Simplify and combine like terms.

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Step 1.18.3.1.2.1

Simplify each term.

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Step 1.18.3.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{600 (2 x \cdot x + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.2.1.2

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

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Step 1.18.3.1.2.1.2.1

Move $x$ .

$\frac{600 (2 (x \cdot x) + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{600 (2 x^{2} + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.2.1.3

Move $- 5$ to the left of $x$ .

$\frac{600 (2 x^{2} - 5 \cdot x + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.2.1.4

Multiply $2 x$ by $1$ .

$\frac{600 (2 x^{2} - 5 x + 2 x + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.2.1.5

Multiply $- 5$ by $1$ .

$\frac{600 (2 x^{2} - 5 x + 2 x - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.2.2

Add $- 5 x$ and $2 x$ .

$\frac{600 (2 x^{2} - 3 x - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.3

Apply the distributive property.

$\frac{600 (2 x^{2}) + 600 (- 3 x) + 600 \cdot - 5 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.4

Simplify.

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Step 1.18.3.1.4.1

Multiply $2$ by $600$ .

$\frac{1200 x^{2} + 600 (- 3 x) + 600 \cdot - 5 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.4.2

Multiply $- 3$ by $600$ .

$\frac{1200 x^{2} - 1800 x + 600 \cdot - 5 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.4.3

Multiply $600$ by $- 5$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.5

Expand $(x - 6) (x + 1)$ using the FOIL Method.

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Step 1.18.3.1.5.1

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x (x + 1) - 6 (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.5.2

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x \cdot x + x \cdot 1 - 6 (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.5.3

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x \cdot x + x \cdot 1 - 6 x - 6 \cdot 1)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.6

Simplify and combine like terms.

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Step 1.18.3.1.6.1

Simplify each term.

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Step 1.18.3.1.6.1.1

Multiply $x$ by $x$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} + x \cdot 1 - 6 x - 6 \cdot 1)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.6.1.2

Multiply $x$ by $1$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} + x - 6 x - 6 \cdot 1)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.6.1.3

Multiply $- 6$ by $1$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} + x - 6 x - 6)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.6.2

Subtract $6 x$ from $x$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} - 5 x - 6)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.7

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2} - 2 (- 5 x) - 2 \cdot - 6) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.8

Simplify.

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Step 1.18.3.1.8.1

Multiply $- 5$ by $- 2$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2} + 10 x - 2 \cdot - 6) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.8.2

Multiply $- 2$ by $- 6$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2} + 10 x + 12) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.9

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2}) + 600 (10 x) + 600 \cdot 12 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.10

Simplify.

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Step 1.18.3.1.10.1

Multiply $- 2$ by $600$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 600 (10 x) + 600 \cdot 12 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.10.2

Multiply $10$ by $600$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 600 \cdot 12 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.10.3

Multiply $600$ by $12$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 1.18.3.1.11

Multiply $600 (- 2 \cdot 14)$ .

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Step 1.18.3.1.11.1

Multiply $- 2$ by $14$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 + 600 \cdot - 28}{{(x + 1)}^{3}}$

Step 1.18.3.1.11.2

Multiply $600$ by $- 28$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 - 16800}{{(x + 1)}^{3}}$

Step 1.18.3.2

Combine the opposite terms in $1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 - 16800$ .

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Step 1.18.3.2.1

Subtract $1200 x^{2}$ from $1200 x^{2}$ .

$\frac{- 1800 x - 3000 + 0 + 6000 x + 7200 - 16800}{{(x + 1)}^{3}}$

Step 1.18.3.2.2

Add $- 1800 x - 3000$ and $0$ .

$\frac{- 1800 x - 3000 + 6000 x + 7200 - 16800}{{(x + 1)}^{3}}$

Step 1.18.3.3

Add $- 1800 x$ and $6000 x$ .

$\frac{4200 x - 3000 + 7200 - 16800}{{(x + 1)}^{3}}$

Step 1.18.3.4

Add $- 3000$ and $7200$ .

$\frac{4200 x + 4200 - 16800}{{(x + 1)}^{3}}$

Step 1.18.3.5

Subtract $16800$ from $4200$ .

$\frac{4200 x - 12600}{{(x + 1)}^{3}}$

Step 1.18.4

Factor $4200$ out of $4200 x - 12600$ .

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Step 1.18.4.1

Factor $4200$ out of $4200 x$ .

$\frac{4200 (x) - 12600}{{(x + 1)}^{3}}$

Step 1.18.4.2

Factor $4200$ out of $- 12600$ .

$\frac{4200 x + 4200 \cdot - 3}{{(x + 1)}^{3}}$

Step 1.18.4.3

Factor $4200$ out of $4200 x + 4200 \cdot - 3$ .

$\frac{4200 (x - 3)}{{(x + 1)}^{3}}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $4200$ is constant with respect to $x$ , the derivative of $\frac{4200 (x - 3)}{{(x + 1)}^{3}}$ with respect to $x$ is $4200 \frac{d}{d x} [\frac{x - 3}{{(x + 1)}^{3}}]$ .

$f'' (x) = 4200 \frac{d}{d x} (\frac{x - 3}{{(x + 1)}^{3}})$

Step 2.2

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} \frac{d}{d x} (x - 3) - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{({(x + 1)}^{3})}^{2}})$

Step 2.3

Differentiate.

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Step 2.3.1

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

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Step 2.3.1.1

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

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} \frac{d}{d x} (x - 3) - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{(x + 1)}^{3 \cdot 2}})$

Step 2.3.1.2

Multiply $3$ by $2$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} \frac{d}{d x} (x - 3) - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{(x + 1)}^{6}})$

Step 2.3.2

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

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} (\frac{d}{d x} (x) + \frac{d}{d x} (- 3)) - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{(x + 1)}^{6}})$

Step 2.3.3

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

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} (1 + \frac{d}{d x} (- 3)) - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{(x + 1)}^{6}})$

Step 2.3.4

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

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} (1 + 0) - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{(x + 1)}^{6}})$

Step 2.3.5

Simplify the expression.

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Step 2.3.5.1

Add $1$ and $0$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} \cdot 1 - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{(x + 1)}^{6}})$

Step 2.3.5.2

Multiply ${(x + 1)}^{3}$ by $1$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} - (x - 3) \frac{d}{d x} {(x + 1)}^{3}}{{(x + 1)}^{6}})$

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 $f (x) = x^{3}$ and $g (x) = x + 1$ .

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Step 2.4.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} - (x - 3) (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x + 1))}{{(x + 1)}^{6}})$

Step 2.4.2

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

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} - (x - 3) (3 u^{2} \frac{d}{d x} (x + 1))}{{(x + 1)}^{6}})$

Step 2.4.3

Replace all occurrences of $u$ with $x + 1$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} - (x - 3) (3 {(x + 1)}^{2} \frac{d}{d x} (x + 1))}{{(x + 1)}^{6}})$

Step 2.5

Simplify with factoring out.

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Step 2.5.1

Multiply $3$ by $- 1$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{3} - 3 (x - 3) ({(x + 1)}^{2} \frac{d}{d x} (x + 1))}{{(x + 1)}^{6}})$

Step 2.5.2

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{3} - 3 (x - 3) ({(x + 1)}^{2} \frac{d}{d x} [x + 1])$ .

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Step 2.5.2.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{3}$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{2} (x + 1) - 3 (x - 3) ({(x + 1)}^{2} \frac{d}{d x} (x + 1))}{{(x + 1)}^{6}})$

Step 2.5.2.2

Factor ${(x + 1)}^{2}$ out of $- 3 (x - 3) ({(x + 1)}^{2} \frac{d}{d x} [x + 1])$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{2} (x + 1) + {(x + 1)}^{2} (- 3 (x - 3) (\frac{d}{d x} (x + 1)))}{{(x + 1)}^{6}})$

Step 2.5.2.3

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{2} (x + 1) + {(x + 1)}^{2} (- 3 (x - 3) (\frac{d}{d x} [x + 1]))$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{2} (x + 1 - 3 (x - 3) (\frac{d}{d x} (x + 1)))}{{(x + 1)}^{6}})$

Step 2.6

Cancel the common factors.

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Step 2.6.1

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{6}$ .

$f'' (x) = 4200 (\frac{{(x + 1)}^{2} (x + 1 - 3 (x - 3) (\frac{d}{d x} (x + 1)))}{{(x + 1)}^{2} {(x + 1)}^{4}})$

Step 2.6.2

Cancel the common factor.

$f'' (x) = 4200 (\frac{{(x + 1)}^{2} (x + 1 - 3 (x - 3) (\frac{d}{d x} (x + 1)))}{{(x + 1)}^{2} {(x + 1)}^{4}})$

Step 2.6.3

Rewrite the expression.

$f'' (x) = 4200 (\frac{x + 1 - 3 (x - 3) (\frac{d}{d x} (x + 1))}{{(x + 1)}^{4}})$

Step 2.7

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

$f'' (x) = 4200 (\frac{x + 1 - 3 (x - 3) (\frac{d}{d x} (x) + \frac{d}{d x} (1))}{{(x + 1)}^{4}})$

Step 2.8

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

$f'' (x) = 4200 (\frac{x + 1 - 3 (x - 3) (1 + \frac{d}{d x} (1))}{{(x + 1)}^{4}})$

Step 2.9

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

$f'' (x) = 4200 (\frac{x + 1 - 3 (x - 3) (1 + 0)}{{(x + 1)}^{4}})$

Step 2.10

Combine fractions.

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Step 2.10.1

Add $1$ and $0$ .

$f'' (x) = 4200 (\frac{x + 1 - 3 (x - 3) \cdot 1}{{(x + 1)}^{4}})$

Step 2.10.2

Multiply $- 3$ by $1$ .

$f'' (x) = 4200 (\frac{x + 1 - 3 (x - 3)}{{(x + 1)}^{4}})$

Step 2.10.3

Combine $4200$ and $\frac{x + 1 - 3 (x - 3)}{{(x + 1)}^{4}}$ .

$f'' (x) = \frac{4200 (x + 1 - 3 (x - 3))}{{(x + 1)}^{4}}$

Step 2.11

Simplify.

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Step 2.11.1

Apply the distributive property.

$f'' (x) = \frac{4200 (x + 1 - 3 x - 3 \cdot - 3)}{{(x + 1)}^{4}}$

Step 2.11.2

Apply the distributive property.

$f'' (x) = \frac{4200 x + 4200 \cdot 1 + 4200 (- 3 x) + 4200 (- 3 \cdot - 3)}{{(x + 1)}^{4}}$

Step 2.11.3

Simplify the numerator.

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Step 2.11.3.1

Simplify each term.

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Step 2.11.3.1.1

Multiply $4200$ by $1$ .

$f'' (x) = \frac{4200 x + 4200 + 4200 (- 3 x) + 4200 (- 3 \cdot - 3)}{{(x + 1)}^{4}}$

Step 2.11.3.1.2

Multiply $- 3$ by $4200$ .

$f'' (x) = \frac{4200 x + 4200 - 12600 x + 4200 (- 3 \cdot - 3)}{{(x + 1)}^{4}}$

Step 2.11.3.1.3

Multiply $4200 (- 3 \cdot - 3)$ .

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Step 2.11.3.1.3.1

Multiply $- 3$ by $- 3$ .

$f'' (x) = \frac{4200 x + 4200 - 12600 x + 4200 \cdot 9}{{(x + 1)}^{4}}$

Step 2.11.3.1.3.2

Multiply $4200$ by $9$ .

$f'' (x) = \frac{4200 x + 4200 - 12600 x + 37800}{{(x + 1)}^{4}}$

Step 2.11.3.2

Subtract $12600 x$ from $4200 x$ .

$f'' (x) = \frac{- 8400 x + 4200 + 37800}{{(x + 1)}^{4}}$

Step 2.11.3.3

Add $4200$ and $37800$ .

$f'' (x) = \frac{- 8400 x + 42000}{{(x + 1)}^{4}}$

Step 2.11.4

Factor $8400$ out of $- 8400 x + 42000$ .

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Step 2.11.4.1

Factor $8400$ out of $- 8400 x$ .

$f'' (x) = \frac{8400 (- x) + 42000}{{(x + 1)}^{4}}$

Step 2.11.4.2

Factor $8400$ out of $42000$ .

$f'' (x) = \frac{8400 (- x) + 8400 (5)}{{(x + 1)}^{4}}$

Step 2.11.4.3

Factor $8400$ out of $8400 (- x) + 8400 (5)$ .

$f'' (x) = \frac{8400 (- x + 5)}{{(x + 1)}^{4}}$

Step 2.11.5

Factor $- 1$ out of $- x$ .

$f'' (x) = \frac{8400 (- (x) + 5)}{{(x + 1)}^{4}}$

Step 2.11.6

Rewrite $5$ as $- 1 (- 5)$ .

$f'' (x) = \frac{8400 (- (x) - 1 \cdot - 5)}{{(x + 1)}^{4}}$

Step 2.11.7

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$f'' (x) = \frac{8400 (- (x - 5))}{{(x + 1)}^{4}}$

Step 2.11.8

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$f'' (x) = \frac{8400 (- 1 (x - 5))}{{(x + 1)}^{4}}$

Step 2.11.9

Move the negative in front of the fraction.

$f'' (x) = - \frac{8400 (x - 5)}{{(x + 1)}^{4}}$

Step 3

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

$\frac{4200 (x - 3)}{{(x + 1)}^{3}} = 0$

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

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

$\frac{d}{d x} [600 (\frac{x + 1}{x + 1} - \frac{7}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.2

Combine the numerators over the common denominator.

$\frac{d}{d x} [600 (\frac{x + 1 - 7}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.3

Subtract $7$ from $1$ .

$\frac{d}{d x} [600 (\frac{x - 6}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.4

To write $\frac{x - 6}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{d}{d x} [600 (\frac{x - 6}{x + 1} \cdot \frac{x + 1}{x + 1} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.5

Write each expression with a common denominator of ${(x + 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.5.1

Multiply $\frac{x - 6}{x + 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{(x + 1) (x + 1)} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.5.2

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

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{1} (x + 1)} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.5.3

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

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{1} {(x + 1)}^{1}} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.5.4

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

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{1 + 1}} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.5.5

Add $1$ and $1$ .

$\frac{d}{d x} [600 (\frac{(x - 6) (x + 1)}{{(x + 1)}^{2}} + \frac{14}{{(x + 1)}^{2}})]$

Step 4.1.6

Differentiate using the Constant Multiple Rule.

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Step 4.1.6.1

Combine the numerators over the common denominator.

$\frac{d}{d x} [600 \frac{(x - 6) (x + 1) + 14}{{(x + 1)}^{2}}]$

Step 4.1.6.2

Combine $600$ and $\frac{(x - 6) (x + 1) + 14}{{(x + 1)}^{2}}$ .

$\frac{d}{d x} [\frac{600 ((x - 6) (x + 1) + 14)}{{(x + 1)}^{2}}]$

Step 4.1.6.3

$600 \frac{d}{d x} [\frac{(x - 6) (x + 1) + 14}{{(x + 1)}^{2}}]$

Step 4.1.7

$600 \frac{{(x + 1)}^{2} \frac{d}{d x} [(x - 6) (x + 1) + 14] - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{({(x + 1)}^{2})}^{2}}$

Step 4.1.8

Differentiate using the Sum Rule.

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Step 4.1.8.1

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

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Step 4.1.8.1.1

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

$600 \frac{{(x + 1)}^{2} \frac{d}{d x} [(x - 6) (x + 1) + 14] - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{2 \cdot 2}}$

Step 4.1.8.1.2

Multiply $2$ by $2$ .

$600 \frac{{(x + 1)}^{2} \frac{d}{d x} [(x - 6) (x + 1) + 14] - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.8.2

By the Sum Rule, the derivative of $(x - 6) (x + 1) + 14$ with respect to $x$ is $\frac{d}{d x} [(x - 6) (x + 1)] + \frac{d}{d x} [14]$ .

$600 \frac{{(x + 1)}^{2} (\frac{d}{d x} [(x - 6) (x + 1)] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.9

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 $g (x) = x + 1$ .

$600 \frac{{(x + 1)}^{2} ((x - 6) \frac{d}{d x} [x + 1] + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10

Differentiate.

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Step 4.1.10.1

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

$600 \frac{{(x + 1)}^{2} ((x - 6) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.2

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

$600 \frac{{(x + 1)}^{2} ((x - 6) (1 + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.3

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

$600 \frac{{(x + 1)}^{2} ((x - 6) (1 + 0) + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.4

Simplify the expression.

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Step 4.1.10.4.1

Add $1$ and $0$ .

$600 \frac{{(x + 1)}^{2} ((x - 6) \cdot 1 + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.4.2

Multiply $x - 6$ by $1$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) \frac{d}{d x} [x - 6] + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.5

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

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 6]) + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.6

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

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) (1 + \frac{d}{d x} [- 6]) + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.7

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) (1 + 0) + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.8

Simplify by adding terms.

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Step 4.1.10.8.1

Add $1$ and $0$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + (x + 1) \cdot 1 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.8.2

Multiply $x + 1$ by $1$ .

$600 \frac{{(x + 1)}^{2} (x - 6 + x + 1 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.8.3

Add $x$ and $x$ .

$600 \frac{{(x + 1)}^{2} (2 x - 6 + 1 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.8.4

Add $- 6$ and $1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5 + \frac{d}{d x} [14]) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.9

Since $14$ is constant with respect to $x$ , the derivative of $14$ with respect to $x$ is $0$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5 + 0) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.10.10

Add $2 x - 5$ and $0$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) \frac{d}{d x} [{(x + 1)}^{2}]}{{(x + 1)}^{4}}$

Step 4.1.11

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 $g (x) = x + 1$ .

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Step 4.1.11.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 4.1.11.2

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

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) (2 u \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 4.1.11.3

Replace all occurrences of $u$ with $x + 1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - ((x - 6) (x + 1) + 14) (2 (x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 4.1.12

Simplify with factoring out.

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Step 4.1.12.1

Multiply $2$ by $- 1$ .

$600 \frac{{(x + 1)}^{2} (2 x - 5) - 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 4.1.12.2

Factor $x + 1$ out of ${(x + 1)}^{2} (2 x - 5) - 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])$ .

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Step 4.1.12.2.1

Factor $x + 1$ out of ${(x + 1)}^{2} (2 x - 5)$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5)) - 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])}{{(x + 1)}^{4}}$

Step 4.1.12.2.2

Factor $x + 1$ out of $- 2 ((x - 6) (x + 1) + 14) ((x + 1) \frac{d}{d x} [x + 1])$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5)) + (x + 1) (- 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{{(x + 1)}^{4}}$

Step 4.1.12.2.3

Factor $x + 1$ out of $(x + 1) ((x + 1) (2 x - 5)) + (x + 1) (- 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{{(x + 1)}^{4}}$

Step 4.1.13

Cancel the common factors.

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Step 4.1.13.1

Factor $x + 1$ out of ${(x + 1)}^{4}$ .

$600 \frac{(x + 1) ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{(x + 1) {(x + 1)}^{3}}$

Step 4.1.13.2

Cancel the common factor.

$600 \frac{(x + 1) ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1]))}{(x + 1) {(x + 1)}^{3}}$

Step 4.1.13.3

Rewrite the expression.

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x + 1])}{{(x + 1)}^{3}}$

Step 4.1.14

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

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (\frac{d}{d x} [x] + \frac{d}{d x} [1])}{{(x + 1)}^{3}}$

Step 4.1.15

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

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (1 + \frac{d}{d x} [1])}{{(x + 1)}^{3}}$

Step 4.1.16

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

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) (1 + 0)}{{(x + 1)}^{3}}$

Step 4.1.17

Combine fractions.

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Step 4.1.17.1

Add $1$ and $0$ .

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14) \cdot 1}{{(x + 1)}^{3}}$

Step 4.1.17.2

Multiply $- 2$ by $1$ .

$600 \frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14)}{{(x + 1)}^{3}}$

Step 4.1.17.3

Combine $600$ and $\frac{(x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14)}{{(x + 1)}^{3}}$ .

$\frac{600 ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1) + 14))}{{(x + 1)}^{3}}$

Step 4.1.18

Simplify.

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Step 4.1.18.1

Apply the distributive property.

$\frac{600 ((x + 1) (2 x - 5) - 2 ((x - 6) (x + 1)) - 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.2

Apply the distributive property.

$\frac{600 ((x + 1) (2 x - 5)) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3

Simplify the numerator.

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Step 4.1.18.3.1

Simplify each term.

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Step 4.1.18.3.1.1

Expand $(x + 1) (2 x - 5)$ using the FOIL Method.

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Step 4.1.18.3.1.1.1

Apply the distributive property.

$\frac{600 (x (2 x - 5) + 1 (2 x - 5)) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.1.2

Apply the distributive property.

$\frac{600 (x (2 x) + x \cdot - 5 + 1 (2 x - 5)) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.1.3

Apply the distributive property.

$\frac{600 (x (2 x) + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.2

Simplify and combine like terms.

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Step 4.1.18.3.1.2.1

Simplify each term.

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Step 4.1.18.3.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{600 (2 x \cdot x + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.2.1.2

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

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Step 4.1.18.3.1.2.1.2.1

Move $x$ .

$\frac{600 (2 (x \cdot x) + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{600 (2 x^{2} + x \cdot - 5 + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.2.1.3

Move $- 5$ to the left of $x$ .

$\frac{600 (2 x^{2} - 5 \cdot x + 1 (2 x) + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.2.1.4

Multiply $2 x$ by $1$ .

$\frac{600 (2 x^{2} - 5 x + 2 x + 1 \cdot - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.2.1.5

Multiply $- 5$ by $1$ .

$\frac{600 (2 x^{2} - 5 x + 2 x - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.2.2

Add $- 5 x$ and $2 x$ .

$\frac{600 (2 x^{2} - 3 x - 5) + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.3

Apply the distributive property.

$\frac{600 (2 x^{2}) + 600 (- 3 x) + 600 \cdot - 5 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.4

Simplify.

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Step 4.1.18.3.1.4.1

Multiply $2$ by $600$ .

$\frac{1200 x^{2} + 600 (- 3 x) + 600 \cdot - 5 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.4.2

Multiply $- 3$ by $600$ .

$\frac{1200 x^{2} - 1800 x + 600 \cdot - 5 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.4.3

Multiply $600$ by $- 5$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 ((x - 6) (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.5

Expand $(x - 6) (x + 1)$ using the FOIL Method.

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Step 4.1.18.3.1.5.1

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x (x + 1) - 6 (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.5.2

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x \cdot x + x \cdot 1 - 6 (x + 1))) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.5.3

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x \cdot x + x \cdot 1 - 6 x - 6 \cdot 1)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.6

Simplify and combine like terms.

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Step 4.1.18.3.1.6.1

Simplify each term.

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Step 4.1.18.3.1.6.1.1

Multiply $x$ by $x$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} + x \cdot 1 - 6 x - 6 \cdot 1)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.6.1.2

Multiply $x$ by $1$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} + x - 6 x - 6 \cdot 1)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.6.1.3

Multiply $- 6$ by $1$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} + x - 6 x - 6)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.6.2

Subtract $6 x$ from $x$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 (x^{2} - 5 x - 6)) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.7

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2} - 2 (- 5 x) - 2 \cdot - 6) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.8

Simplify.

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Step 4.1.18.3.1.8.1

Multiply $- 5$ by $- 2$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2} + 10 x - 2 \cdot - 6) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.8.2

Multiply $- 2$ by $- 6$ .

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2} + 10 x + 12) + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.9

Apply the distributive property.

$\frac{1200 x^{2} - 1800 x - 3000 + 600 (- 2 x^{2}) + 600 (10 x) + 600 \cdot 12 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.10

Simplify.

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Step 4.1.18.3.1.10.1

Multiply $- 2$ by $600$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 600 (10 x) + 600 \cdot 12 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.10.2

Multiply $10$ by $600$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 600 \cdot 12 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.10.3

Multiply $600$ by $12$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 + 600 (- 2 \cdot 14)}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.11

Multiply $600 (- 2 \cdot 14)$ .

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Step 4.1.18.3.1.11.1

Multiply $- 2$ by $14$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 + 600 \cdot - 28}{{(x + 1)}^{3}}$

Step 4.1.18.3.1.11.2

Multiply $600$ by $- 28$ .

$\frac{1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 - 16800}{{(x + 1)}^{3}}$

Step 4.1.18.3.2

Combine the opposite terms in $1200 x^{2} - 1800 x - 3000 - 1200 x^{2} + 6000 x + 7200 - 16800$ .

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Step 4.1.18.3.2.1

Subtract $1200 x^{2}$ from $1200 x^{2}$ .

$\frac{- 1800 x - 3000 + 0 + 6000 x + 7200 - 16800}{{(x + 1)}^{3}}$

Step 4.1.18.3.2.2

Add $- 1800 x - 3000$ and $0$ .

$\frac{- 1800 x - 3000 + 6000 x + 7200 - 16800}{{(x + 1)}^{3}}$

Step 4.1.18.3.3

Add $- 1800 x$ and $6000 x$ .

$\frac{4200 x - 3000 + 7200 - 16800}{{(x + 1)}^{3}}$

Step 4.1.18.3.4

Add $- 3000$ and $7200$ .

$\frac{4200 x + 4200 - 16800}{{(x + 1)}^{3}}$

Step 4.1.18.3.5

Subtract $16800$ from $4200$ .

$\frac{4200 x - 12600}{{(x + 1)}^{3}}$

Step 4.1.18.4

Factor $4200$ out of $4200 x - 12600$ .

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Step 4.1.18.4.1

Factor $4200$ out of $4200 x$ .

$\frac{4200 (x) - 12600}{{(x + 1)}^{3}}$

Step 4.1.18.4.2

Factor $4200$ out of $- 12600$ .

$\frac{4200 x + 4200 \cdot - 3}{{(x + 1)}^{3}}$

Step 4.1.18.4.3

Factor $4200$ out of $4200 x + 4200 \cdot - 3$ .

$f' (x) = \frac{4200 (x - 3)}{{(x + 1)}^{3}}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{4200 (x - 3)}{{(x + 1)}^{3}}$ .

$\frac{4200 (x - 3)}{{(x + 1)}^{3}}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{4200 (x - 3)}{{(x + 1)}^{3}} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{4200 (x - 3)}{{(x + 1)}^{3}} = 0$

Step 5.2

Set the numerator equal to zero.

$4200 (x - 3) = 0$

Step 5.3

Solve the equation for $x$ .

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Step 5.3.1

Divide each term in $4200 (x - 3) = 0$ by $4200$ and simplify.

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Step 5.3.1.1

Divide each term in $4200 (x - 3) = 0$ by $4200$ .

$\frac{4200 (x - 3)}{4200} = \frac{0}{4200}$

Step 5.3.1.2

Simplify the left side.

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Step 5.3.1.2.1

Cancel the common factor of $4200$ .

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Step 5.3.1.2.1.1

Cancel the common factor.

$\frac{4200 (x - 3)}{4200} = \frac{0}{4200}$

Step 5.3.1.2.1.2

Divide $x - 3$ by $1$ .

$x - 3 = \frac{0}{4200}$

Step 5.3.1.3

Simplify the right side.

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Step 5.3.1.3.1

Divide $0$ by $4200$ .

$x - 3 = 0$

Step 5.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{4200 (x - 3)}{{(x + 1)}^{3}}$ equal to $0$ to find where the expression is undefined.

${(x + 1)}^{3} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 6.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

Critical points to evaluate.

$x = 3$

Step 8

Evaluate the second derivative at $x = 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{8400 ((3) - 5)}{{((3) + 1)}^{4}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Subtract $5$ from $3$ .

$- \frac{8400 \cdot - 2}{{(3 + 1)}^{4}}$

Step 9.2

Simplify the denominator.

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Step 9.2.1

Add $3$ and $1$ .

$- \frac{8400 \cdot - 2}{4^{4}}$

Step 9.2.2

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

$- \frac{8400 \cdot - 2}{256}$

Step 9.3

Reduce the expression by cancelling the common factors.

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Step 9.3.1

Multiply $8400$ by $- 2$ .

$- \frac{- 16800}{256}$

Step 9.3.2

Cancel the common factor of $- 16800$ and $256$ .

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Step 9.3.2.1

Factor $32$ out of $- 16800$ .

$- \frac{32 (- 525)}{256}$

Step 9.3.2.2

Cancel the common factors.

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Step 9.3.2.2.1

Factor $32$ out of $256$ .

$- \frac{32 \cdot - 525}{32 \cdot 8}$

Step 9.3.2.2.2

Cancel the common factor.

$- \frac{32 \cdot - 525}{32 \cdot 8}$

Step 9.3.2.2.3

Rewrite the expression.

$- \frac{- 525}{8}$

Step 9.3.3

Move the negative in front of the fraction.

$- - \frac{525}{8}$

Step 9.4

Multiply $- - \frac{525}{8}$ .

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Step 9.4.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{525}{8})$

Step 9.4.2

Multiply $\frac{525}{8}$ by $1$ .

$\frac{525}{8}$

Step 10

$x = 3$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 3$ is a local minimum

Step 11

Find the y-value when $x = 3$ .

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Step 11.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = 600 (1 - \frac{7}{(3) + 1} + \frac{14}{{((3) + 1)}^{2}})$

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

Add $3$ and $1$ .

$f (3) = 600 (1 - \frac{7}{4} + \frac{14}{{((3) + 1)}^{2}})$

Step 11.2.1.2

Simplify the denominator.

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Step 11.2.1.2.1

Add $3$ and $1$ .

$f (3) = 600 (1 - \frac{7}{4} + \frac{14}{4^{2}})$

Step 11.2.1.2.2

Raise $4$ to the power of $2$ .

$f (3) = 600 (1 - \frac{7}{4} + \frac{14}{16})$

Step 11.2.1.3

Cancel the common factor of $14$ and $16$ .

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Step 11.2.1.3.1

Factor $2$ out of $14$ .

$f (3) = 600 (1 - \frac{7}{4} + \frac{2 (7)}{16})$

Step 11.2.1.3.2

Cancel the common factors.

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Step 11.2.1.3.2.1

Factor $2$ out of $16$ .

$f (3) = 600 (1 - \frac{7}{4} + \frac{2 \cdot 7}{2 \cdot 8})$

Step 11.2.1.3.2.2

Cancel the common factor.

$f (3) = 600 (1 - \frac{7}{4} + \frac{2 \cdot 7}{2 \cdot 8})$

Step 11.2.1.3.2.3

Rewrite the expression.

$f (3) = 600 (1 - \frac{7}{4} + \frac{7}{8})$

Step 11.2.2

Find the common denominator.

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Step 11.2.2.1

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

$f (3) = 600 (\frac{1}{1} - \frac{7}{4} + \frac{7}{8})$

Step 11.2.2.2

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$f (3) = 600 (\frac{1}{1} \cdot \frac{8}{8} - \frac{7}{4} + \frac{7}{8})$

Step 11.2.2.3

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$f (3) = 600 (\frac{8}{8} - \frac{7}{4} + \frac{7}{8})$

Step 11.2.2.4

Multiply $\frac{7}{4}$ by $\frac{2}{2}$ .

$f (3) = 600 (\frac{8}{8} - (\frac{7}{4} \cdot \frac{2}{2}) + \frac{7}{8})$

Step 11.2.2.5

Multiply $\frac{7}{4}$ by $\frac{2}{2}$ .

$f (3) = 600 (\frac{8}{8} - \frac{7 \cdot 2}{4 \cdot 2} + \frac{7}{8})$

Step 11.2.2.6

Reorder the factors of $4 \cdot 2$ .

$f (3) = 600 (\frac{8}{8} - \frac{7 \cdot 2}{2 \cdot 4} + \frac{7}{8})$

Step 11.2.2.7

Multiply $2$ by $4$ .

$f (3) = 600 (\frac{8}{8} - \frac{7 \cdot 2}{8} + \frac{7}{8})$

Step 11.2.3

Combine the numerators over the common denominator.

$f (3) = 600 (\frac{8 - 7 \cdot 2 + 7}{8})$

Step 11.2.4

Simplify the expression.

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Step 11.2.4.1

Multiply $- 7$ by $2$ .

$f (3) = 600 (\frac{8 - 14 + 7}{8})$

Step 11.2.4.2

Subtract $14$ from $8$ .

$f (3) = 600 (\frac{- 6 + 7}{8})$

Step 11.2.4.3

Add $- 6$ and $7$ .

$f (3) = 600 (\frac{1}{8})$

Step 11.2.5

Cancel the common factor of $8$ .

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Step 11.2.5.1

Factor $8$ out of $600$ .

$f (3) = 8 (75) (\frac{1}{8})$

Step 11.2.5.2

Cancel the common factor.

$f (3) = 8 \cdot (75 (\frac{1}{8}))$

Step 11.2.5.3

Rewrite the expression.

$f (3) = 75$

Step 11.2.6

The final answer is $75$ .

$y = 75$

Step 12

These are the local extrema for $f (x) = 600 (1 - \frac{7}{x + 1} + \frac{14}{{(x + 1)}^{2}})$ .

$(3, 75)$ is a local minima

Step 13