Calculus Examples

$\frac{4 (12 x^{2} - 16 x - 7)}{{(3 x - 1)}^{2}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4 (12 x^{2} - 16 x - 7)}{{(3 x - 1)}^{2}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{12 x^{2} - 16 x - 7}{{(3 x - 1)}^{2}}]$ .

$4 \frac{d}{d x} [\frac{12 x^{2} - 16 x - 7}{{(3 x - 1)}^{2}}]$

Step 1.1.2

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 $f (x) = 12 x^{2} - 16 x - 7$ and $g (x) = {(3 x - 1)}^{2}$ .

$4 \frac{{(3 x - 1)}^{2} \frac{d}{d x} [12 x^{2} - 16 x - 7] - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{({(3 x - 1)}^{2})}^{2}}$

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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Step 1.1.3.1.1

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

$4 \frac{{(3 x - 1)}^{2} \frac{d}{d x} [12 x^{2} - 16 x - 7] - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{2 \cdot 2}}$

Step 1.1.3.1.2

Multiply $2$ by $2$ .

$4 \frac{{(3 x - 1)}^{2} \frac{d}{d x} [12 x^{2} - 16 x - 7] - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.2

By the Sum Rule, the derivative of $12 x^{2} - 16 x - 7$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 16 x] + \frac{d}{d x} [- 7]$ .

$4 \frac{{(3 x - 1)}^{2} (\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 16 x] + \frac{d}{d x} [- 7]) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.3

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$4 \frac{{(3 x - 1)}^{2} (12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 16 x] + \frac{d}{d x} [- 7]) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.4

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

$4 \frac{{(3 x - 1)}^{2} (12 (2 x) + \frac{d}{d x} [- 16 x] + \frac{d}{d x} [- 7]) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.5

Multiply $2$ by $12$ .

$4 \frac{{(3 x - 1)}^{2} (24 x + \frac{d}{d x} [- 16 x] + \frac{d}{d x} [- 7]) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.6

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x$ with respect to $x$ is $- 16 \frac{d}{d x} [x]$ .

$4 \frac{{(3 x - 1)}^{2} (24 x - 16 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.7

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

$4 \frac{{(3 x - 1)}^{2} (24 x - 16 \cdot 1 + \frac{d}{d x} [- 7]) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.8

Multiply $- 16$ by $1$ .

$4 \frac{{(3 x - 1)}^{2} (24 x - 16 + \frac{d}{d x} [- 7]) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.9

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$4 \frac{{(3 x - 1)}^{2} (24 x - 16 + 0) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.3.10

Add $24 x - 16$ and $0$ .

$4 \frac{{(3 x - 1)}^{2} (24 x - 16) - (12 x^{2} - 16 x - 7) \frac{d}{d x} [{(3 x - 1)}^{2}]}{{(3 x - 1)}^{4}}$

Step 1.1.4

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

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Step 1.1.4.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

$4 \frac{{(3 x - 1)}^{2} (24 x - 16) - (12 x^{2} - 16 x - 7) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [3 x - 1])}{{(3 x - 1)}^{4}}$

Step 1.1.4.2

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

$4 \frac{{(3 x - 1)}^{2} (24 x - 16) - (12 x^{2} - 16 x - 7) (2 u \frac{d}{d x} [3 x - 1])}{{(3 x - 1)}^{4}}$

Step 1.1.4.3

Replace all occurrences of $u$ with $3 x - 1$ .

$4 \frac{{(3 x - 1)}^{2} (24 x - 16) - (12 x^{2} - 16 x - 7) (2 (3 x - 1) \frac{d}{d x} [3 x - 1])}{{(3 x - 1)}^{4}}$

Step 1.1.5

Simplify with factoring out.

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Step 1.1.5.1

Multiply $2$ by $- 1$ .

$4 \frac{{(3 x - 1)}^{2} (24 x - 16) - 2 (12 x^{2} - 16 x - 7) ((3 x - 1) \frac{d}{d x} [3 x - 1])}{{(3 x - 1)}^{4}}$

Step 1.1.5.2

Factor $3 x - 1$ out of ${(3 x - 1)}^{2} (24 x - 16) - 2 (12 x^{2} - 16 x - 7) ((3 x - 1) \frac{d}{d x} [3 x - 1])$ .

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Step 1.1.5.2.1

Factor $3 x - 1$ out of ${(3 x - 1)}^{2} (24 x - 16)$ .

$4 \frac{(3 x - 1) ((3 x - 1) (24 x - 16)) - 2 (12 x^{2} - 16 x - 7) ((3 x - 1) \frac{d}{d x} [3 x - 1])}{{(3 x - 1)}^{4}}$

Step 1.1.5.2.2

Factor $3 x - 1$ out of $- 2 (12 x^{2} - 16 x - 7) ((3 x - 1) \frac{d}{d x} [3 x - 1])$ .

$4 \frac{(3 x - 1) ((3 x - 1) (24 x - 16)) + (3 x - 1) (- 2 (12 x^{2} - 16 x - 7) (\frac{d}{d x} [3 x - 1]))}{{(3 x - 1)}^{4}}$

Step 1.1.5.2.3

Factor $3 x - 1$ out of $(3 x - 1) ((3 x - 1) (24 x - 16)) + (3 x - 1) (- 2 (12 x^{2} - 16 x - 7) (\frac{d}{d x} [3 x - 1]))$ .

$4 \frac{(3 x - 1) ((3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (\frac{d}{d x} [3 x - 1]))}{{(3 x - 1)}^{4}}$

Step 1.1.6

Cancel the common factors.

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Step 1.1.6.1

Factor $3 x - 1$ out of ${(3 x - 1)}^{4}$ .

$4 \frac{(3 x - 1) ((3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (\frac{d}{d x} [3 x - 1]))}{(3 x - 1) {(3 x - 1)}^{3}}$

Step 1.1.6.2

Cancel the common factor.

$4 \frac{(3 x - 1) ((3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (\frac{d}{d x} [3 x - 1]))}{(3 x - 1) {(3 x - 1)}^{3}}$

Step 1.1.6.3

Rewrite the expression.

$4 \frac{(3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (\frac{d}{d x} [3 x - 1])}{{(3 x - 1)}^{3}}$

Step 1.1.7

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

$4 \frac{(3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{3}}$

Step 1.1.8

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

$4 \frac{(3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{3}}$

Step 1.1.9

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

$4 \frac{(3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (3 \cdot 1 + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{3}}$

Step 1.1.10

Multiply $3$ by $1$ .

$4 \frac{(3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (3 + \frac{d}{d x} [- 1])}{{(3 x - 1)}^{3}}$

Step 1.1.11

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

$4 \frac{(3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) (3 + 0)}{{(3 x - 1)}^{3}}$

Step 1.1.12

Combine fractions.

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Step 1.1.12.1

Add $3$ and $0$ .

$4 \frac{(3 x - 1) (24 x - 16) - 2 (12 x^{2} - 16 x - 7) \cdot 3}{{(3 x - 1)}^{3}}$

Step 1.1.12.2

Multiply $3$ by $- 2$ .

$4 \frac{(3 x - 1) (24 x - 16) - 6 (12 x^{2} - 16 x - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.12.3

Combine $4$ and $\frac{(3 x - 1) (24 x - 16) - 6 (12 x^{2} - 16 x - 7)}{{(3 x - 1)}^{3}}$ .

$\frac{4 ((3 x - 1) (24 x - 16) - 6 (12 x^{2} - 16 x - 7))}{{(3 x - 1)}^{3}}$

Step 1.1.13

Simplify.

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Step 1.1.13.1

Apply the distributive property.

$\frac{4 ((3 x - 1) (24 x - 16) - 6 (12 x^{2}) - 6 (- 16 x) - 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.2

Apply the distributive property.

$\frac{4 ((3 x - 1) (24 x - 16)) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3

Simplify the numerator.

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Step 1.1.13.3.1

Simplify each term.

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Step 1.1.13.3.1.1

Expand $(3 x - 1) (24 x - 16)$ using the FOIL Method.

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Step 1.1.13.3.1.1.1

Apply the distributive property.

$\frac{4 (3 x (24 x - 16) - 1 (24 x - 16)) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.1.2

Apply the distributive property.

$\frac{4 (3 x (24 x) + 3 x \cdot - 16 - 1 (24 x - 16)) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.1.3

Apply the distributive property.

$\frac{4 (3 x (24 x) + 3 x \cdot - 16 - 1 (24 x) - 1 \cdot - 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2

Simplify and combine like terms.

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Step 1.1.13.3.1.2.1

Simplify each term.

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Step 1.1.13.3.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{4 (3 \cdot 24 x \cdot x + 3 x \cdot - 16 - 1 (24 x) - 1 \cdot - 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2.1.2

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

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Step 1.1.13.3.1.2.1.2.1

Move $x$ .

$\frac{4 (3 \cdot 24 (x \cdot x) + 3 x \cdot - 16 - 1 (24 x) - 1 \cdot - 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{4 (3 \cdot 24 x^{2} + 3 x \cdot - 16 - 1 (24 x) - 1 \cdot - 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2.1.3

Multiply $3$ by $24$ .

$\frac{4 (72 x^{2} + 3 x \cdot - 16 - 1 (24 x) - 1 \cdot - 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2.1.4

Multiply $- 16$ by $3$ .

$\frac{4 (72 x^{2} - 48 x - 1 (24 x) - 1 \cdot - 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2.1.5

Multiply $24$ by $- 1$ .

$\frac{4 (72 x^{2} - 48 x - 24 x - 1 \cdot - 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2.1.6

Multiply $- 1$ by $- 16$ .

$\frac{4 (72 x^{2} - 48 x - 24 x + 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.2.2

Subtract $24 x$ from $- 48 x$ .

$\frac{4 (72 x^{2} - 72 x + 16) + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.3

Apply the distributive property.

$\frac{4 (72 x^{2}) + 4 (- 72 x) + 4 \cdot 16 + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.4

Simplify.

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Step 1.1.13.3.1.4.1

Multiply $72$ by $4$ .

$\frac{288 x^{2} + 4 (- 72 x) + 4 \cdot 16 + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.4.2

Multiply $- 72$ by $4$ .

$\frac{288 x^{2} - 288 x + 4 \cdot 16 + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.4.3

Multiply $4$ by $16$ .

$\frac{288 x^{2} - 288 x + 64 + 4 (- 6 (12 x^{2})) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.5

Multiply $12$ by $- 6$ .

$\frac{288 x^{2} - 288 x + 64 + 4 (- 72 x^{2}) + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.6

Multiply $- 72$ by $4$ .

$\frac{288 x^{2} - 288 x + 64 - 288 x^{2} + 4 (- 6 (- 16 x)) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.7

Multiply $- 16$ by $- 6$ .

$\frac{288 x^{2} - 288 x + 64 - 288 x^{2} + 4 (96 x) + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.8

Multiply $96$ by $4$ .

$\frac{288 x^{2} - 288 x + 64 - 288 x^{2} + 384 x + 4 (- 6 \cdot - 7)}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.9

Multiply $4 (- 6 \cdot - 7)$ .

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Step 1.1.13.3.1.9.1

Multiply $- 6$ by $- 7$ .

$\frac{288 x^{2} - 288 x + 64 - 288 x^{2} + 384 x + 4 \cdot 42}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.1.9.2

Multiply $4$ by $42$ .

$\frac{288 x^{2} - 288 x + 64 - 288 x^{2} + 384 x + 168}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.2

Combine the opposite terms in $288 x^{2} - 288 x + 64 - 288 x^{2} + 384 x + 168$ .

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Step 1.1.13.3.2.1

Subtract $288 x^{2}$ from $288 x^{2}$ .

$\frac{- 288 x + 64 + 0 + 384 x + 168}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.2.2

Add $- 288 x + 64$ and $0$ .

$\frac{- 288 x + 64 + 384 x + 168}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.3

Add $- 288 x$ and $384 x$ .

$\frac{96 x + 64 + 168}{{(3 x - 1)}^{3}}$

Step 1.1.13.3.4

Add $64$ and $168$ .

$\frac{96 x + 232}{{(3 x - 1)}^{3}}$

Step 1.1.13.4

Factor $8$ out of $96 x + 232$ .

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Step 1.1.13.4.1

Factor $8$ out of $96 x$ .

$\frac{8 (12 x) + 232}{{(3 x - 1)}^{3}}$

Step 1.1.13.4.2

Factor $8$ out of $232$ .

$\frac{8 (12 x) + 8 (29)}{{(3 x - 1)}^{3}}$

Step 1.1.13.4.3

Factor $8$ out of $8 (12 x) + 8 (29)$ .

$f' (x) = \frac{8 (12 x + 29)}{{(3 x - 1)}^{3}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{8 (12 x + 29)}{{(3 x - 1)}^{3}}$ .

$\frac{8 (12 x + 29)}{{(3 x - 1)}^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{8 (12 x + 29)}{{(3 x - 1)}^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{8 (12 x + 29)}{{(3 x - 1)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$8 (12 x + 29) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $8 (12 x + 29) = 0$ by $8$ and simplify.

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Step 2.3.1.1

Divide each term in $8 (12 x + 29) = 0$ by $8$ .

$\frac{8 (12 x + 29)}{8} = \frac{0}{8}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $8$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{8 (12 x + 29)}{8} = \frac{0}{8}$

Step 2.3.1.2.1.2

Divide $12 x + 29$ by $1$ .

$12 x + 29 = \frac{0}{8}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $8$ .

$12 x + 29 = 0$

Step 2.3.2

Subtract $29$ from both sides of the equation.

$12 x = - 29$

Step 2.3.3

Divide each term in $12 x = - 29$ by $12$ and simplify.

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Step 2.3.3.1

Divide each term in $12 x = - 29$ by $12$ .

$\frac{12 x}{12} = \frac{- 29}{12}$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- 29}{12}$

Step 2.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 29}{12}$

Step 2.3.3.3

Simplify the right side.

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Step 2.3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{29}{12}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{8 (12 x + 29)}{{(3 x - 1)}^{3}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 3.2.2

Solve for $x$ .

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Step 3.2.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 3.2.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 3.2.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.2.2.2

Simplify the left side.

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Step 3.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 3.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 4

Evaluate $\frac{4 (12 x^{2} - 16 x - 7)}{{(3 x - 1)}^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{29}{12}$ .

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Step 4.1.1

Substitute $- \frac{29}{12}$ for $x$ .

$\frac{4 (12 {(- \frac{29}{12})}^{2} - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.1.1

Apply the product rule to $- \frac{29}{12}$ .

$\frac{4 (12 ({(- 1)}^{2} {(\frac{29}{12})}^{2}) - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.1.2

Apply the product rule to $\frac{29}{12}$ .

$\frac{4 (12 ({(- 1)}^{2} \frac{29^{2}}{12^{2}}) - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.2

Raise $- 1$ to the power of $2$ .

$\frac{4 (12 (1 \frac{29^{2}}{12^{2}}) - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.3

Multiply $\frac{29^{2}}{12^{2}}$ by $1$ .

$\frac{4 (12 \frac{29^{2}}{12^{2}} - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.4

Raise $29$ to the power of $2$ .

$\frac{4 (12 \frac{841}{12^{2}} - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.5

Raise $12$ to the power of $2$ .

$\frac{4 (12 (\frac{841}{144}) - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.6

Cancel the common factor of $12$ .

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Step 4.1.2.1.6.1

Factor $12$ out of $144$ .

$\frac{4 (12 \frac{841}{12 (12)} - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.6.2

Cancel the common factor.

$\frac{4 (12 \frac{841}{12 \cdot 12} - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.6.3

Rewrite the expression.

$\frac{4 (\frac{841}{12} - 16 (- \frac{29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.7

Cancel the common factor of $4$ .

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Step 4.1.2.1.7.1

Move the leading negative in $- \frac{29}{12}$ into the numerator.

$\frac{4 (\frac{841}{12} - 16 (\frac{- 29}{12}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.7.2

Factor $4$ out of $- 16$ .

$\frac{4 (\frac{841}{12} + 4 (- 4) \frac{- 29}{12} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.7.3

Factor $4$ out of $12$ .

$\frac{4 (\frac{841}{12} + 4 \cdot - 4 \frac{- 29}{4 \cdot 3} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.7.4

Cancel the common factor.

$\frac{4 (\frac{841}{12} + 4 \cdot - 4 \frac{- 29}{4 \cdot 3} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.7.5

Rewrite the expression.

$\frac{4 (\frac{841}{12} - 4 (\frac{- 29}{3}) - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.8

Combine $- 4$ and $\frac{- 29}{3}$ .

$\frac{4 (\frac{841}{12} + \frac{- 4 \cdot - 29}{3} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.9

Multiply $- 4$ by $- 29$ .

$\frac{4 (\frac{841}{12} + \frac{116}{3} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.10

To write $\frac{116}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4 (\frac{841}{12} + \frac{116}{3} \cdot \frac{4}{4} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.11

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.1.11.1

Multiply $\frac{116}{3}$ by $\frac{4}{4}$ .

$\frac{4 (\frac{841}{12} + \frac{116 \cdot 4}{3 \cdot 4} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.11.2

Multiply $3$ by $4$ .

$\frac{4 (\frac{841}{12} + \frac{116 \cdot 4}{12} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.12

Combine the numerators over the common denominator.

$\frac{4 (\frac{841 + 116 \cdot 4}{12} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.13

Simplify the numerator.

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Step 4.1.2.1.13.1

Multiply $116$ by $4$ .

$\frac{4 (\frac{841 + 464}{12} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.13.2

Add $841$ and $464$ .

$\frac{4 (\frac{1305}{12} - 7)}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.14

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{4 (\frac{1305}{12} - 7 \cdot \frac{12}{12})}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.15

Combine $- 7$ and $\frac{12}{12}$ .

$\frac{4 (\frac{1305}{12} + \frac{- 7 \cdot 12}{12})}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.16

Combine the numerators over the common denominator.

$\frac{4 \frac{1305 - 7 \cdot 12}{12}}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.17

Simplify the numerator.

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Step 4.1.2.1.17.1

Multiply $- 7$ by $12$ .

$\frac{4 \frac{1305 - 84}{12}}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.17.2

Subtract $84$ from $1305$ .

$\frac{4 (\frac{1221}{12})}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.18

Cancel the common factor of $1221$ and $12$ .

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Step 4.1.2.1.18.1

Factor $3$ out of $1221$ .

$\frac{4 \frac{3 (407)}{12}}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.18.2

Cancel the common factors.

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Step 4.1.2.1.18.2.1

Factor $3$ out of $12$ .

$\frac{4 \frac{3 \cdot 407}{3 \cdot 4}}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.18.2.2

Cancel the common factor.

$\frac{4 \frac{3 \cdot 407}{3 \cdot 4}}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.1.18.2.3

Rewrite the expression.

$\frac{4 (\frac{407}{4})}{{(3 (- \frac{29}{12}) - 1)}^{2}}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Cancel the common factor of $3$ .

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Step 4.1.2.2.1.1

Move the leading negative in $- \frac{29}{12}$ into the numerator.

$\frac{4 (\frac{407}{4})}{{(3 (\frac{- 29}{12}) - 1)}^{2}}$

Step 4.1.2.2.1.2

Factor $3$ out of $12$ .

$\frac{4 (\frac{407}{4})}{{(3 \frac{- 29}{3 (4)} - 1)}^{2}}$

Step 4.1.2.2.1.3

Cancel the common factor.

$\frac{4 (\frac{407}{4})}{{(3 \frac{- 29}{3 \cdot 4} - 1)}^{2}}$

Step 4.1.2.2.1.4

Rewrite the expression.

$\frac{4 (\frac{407}{4})}{{(\frac{- 29}{4} - 1)}^{2}}$

Step 4.1.2.2.2

Move the negative in front of the fraction.

$\frac{4 (\frac{407}{4})}{{(- \frac{29}{4} - 1)}^{2}}$

Step 4.1.2.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{4 (\frac{407}{4})}{{(- \frac{29}{4} - 1 \cdot \frac{4}{4})}^{2}}$

Step 4.1.2.2.4

Combine $- 1$ and $\frac{4}{4}$ .

$\frac{4 (\frac{407}{4})}{{(- \frac{29}{4} + \frac{- 1 \cdot 4}{4})}^{2}}$

Step 4.1.2.2.5

Combine the numerators over the common denominator.

$\frac{4 (\frac{407}{4})}{{(\frac{- 29 - 1 \cdot 4}{4})}^{2}}$

Step 4.1.2.2.6

Simplify the numerator.

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Step 4.1.2.2.6.1

Multiply $- 1$ by $4$ .

$\frac{4 (\frac{407}{4})}{{(\frac{- 29 - 4}{4})}^{2}}$

Step 4.1.2.2.6.2

Subtract $4$ from $- 29$ .

$\frac{4 (\frac{407}{4})}{{(\frac{- 33}{4})}^{2}}$

Step 4.1.2.2.7

Move the negative in front of the fraction.

$\frac{4 (\frac{407}{4})}{{(- \frac{33}{4})}^{2}}$

Step 4.1.2.2.8

Apply the product rule to $- \frac{33}{4}$ .

$\frac{4 (\frac{407}{4})}{{(- 1)}^{2} {(\frac{33}{4})}^{2}}$

Step 4.1.2.2.9

Raise $- 1$ to the power of $2$ .

$\frac{4 (\frac{407}{4})}{1 {(\frac{33}{4})}^{2}}$

Step 4.1.2.2.10

Apply the product rule to $\frac{33}{4}$ .

$\frac{4 (\frac{407}{4})}{1 \frac{33^{2}}{4^{2}}}$

Step 4.1.2.2.11

Raise $33$ to the power of $2$ .

$\frac{4 (\frac{407}{4})}{1 \frac{1089}{4^{2}}}$

Step 4.1.2.2.12

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

$\frac{4 (\frac{407}{4})}{1 (\frac{1089}{16})}$

Step 4.1.2.2.13

Multiply $\frac{1089}{16}$ by $1$ .

$\frac{4 (\frac{407}{4})}{\frac{1089}{16}}$

Step 4.1.2.3

Combine fractions.

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Step 4.1.2.3.1

Combine $4$ and $\frac{407}{4}$ .

$\frac{\frac{4 \cdot 407}{4}}{\frac{1089}{16}}$

Step 4.1.2.3.2

Simplify the expression.

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Step 4.1.2.3.2.1

Multiply $4$ by $407$ .

$\frac{\frac{1628}{4}}{\frac{1089}{16}}$

Step 4.1.2.3.2.2

Divide $1628$ by $4$ .

$\frac{407}{\frac{1089}{16}}$

Step 4.1.2.4

Multiply the numerator by the reciprocal of the denominator.

$407 (\frac{16}{1089})$

Step 4.1.2.5

Cancel the common factor of $11$ .

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Step 4.1.2.5.1

Factor $11$ out of $407$ .

$11 (37) \frac{16}{1089}$

Step 4.1.2.5.2

Factor $11$ out of $1089$ .

$11 \cdot 37 \frac{16}{11 \cdot 99}$

Step 4.1.2.5.3

Cancel the common factor.

$11 \cdot 37 \frac{16}{11 \cdot 99}$

Step 4.1.2.5.4

Rewrite the expression.

$37 (\frac{16}{99})$

Step 4.1.2.6

Combine $37$ and $\frac{16}{99}$ .

$\frac{37 \cdot 16}{99}$

Step 4.1.2.7

Multiply $37$ by $16$ .

$\frac{592}{99}$

Step 4.2

Evaluate at $x = \frac{1}{3}$ .

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Step 4.2.1

Substitute $\frac{1}{3}$ for $x$ .

$\frac{4 (12 {(\frac{1}{3})}^{2} - 16 (\frac{1}{3}) - 7)}{{(3 (\frac{1}{3}) - 1)}^{2}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{4 (12 {(\frac{1}{3})}^{2} - 16 (\frac{1}{3}) - 7)}{{(3 (\frac{1}{3}) - 1)}^{2}}$

Step 4.2.2.1.2

Rewrite the expression.

$\frac{4 (12 {(\frac{1}{3})}^{2} - 16 (\frac{1}{3}) - 7)}{{(1 - 1)}^{2}}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Subtract $1$ from $1$ .

$\frac{4 (12 {(\frac{1}{3})}^{2} - 16 (\frac{1}{3}) - 7)}{0^{2}}$

Step 4.2.2.2.2

Raising $0$ to any positive power yields $0$ .

$\frac{4 (12 {(\frac{1}{3})}^{2} - 16 (\frac{1}{3}) - 7)}{0}$

Step 4.2.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{4 (12 {(\frac{1}{3})}^{2} - 16 (\frac{1}{3}) - 7)}{0}$

Step 4.2.2.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 4.3

List all of the points.

$(- \frac{29}{12}, \frac{592}{99})$

Step 5