Calculus Examples

$y = \frac{x^{2} - 9 x}{3 x - x^{3}}$ at $x = 3$

Step 1

Find the corresponding $y$ -value to $x = 3$ .

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Step 1.1

Substitute $3$ in for $x$ .

$y = \frac{{(3)}^{2} - 9 \cdot 3}{3 (3) - {(3)}^{3}}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{3^{2} - 9 \cdot 3}{3 (3) - {(3)}^{3}}$

Step 1.2.2

Remove parentheses.

$y = \frac{3^{2} - 9 \cdot 3}{3 (3) - 3^{3}}$

Step 1.2.3

Remove parentheses.

$y = \frac{{(3)}^{2} - 9 \cdot 3}{3 (3) - {(3)}^{3}}$

Step 1.2.4

Simplify $\frac{{(3)}^{2} - 9 \cdot 3}{3 (3) - {(3)}^{3}}$ .

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Step 1.2.4.1

Cancel the common factor of ${(3)}^{2} - 9 \cdot 3$ and $3 (3) - {(3)}^{3}$ .

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Step 1.2.4.1.1

Reorder terms.

$y = \frac{{(3)}^{2} - 9 \cdot 3}{- {(3)}^{3} + 3 \cdot 3}$

Step 1.2.4.1.2

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

$y = \frac{3 \cdot 3 - 9 \cdot 3}{- {(3)}^{3} + 3 \cdot 3}$

Step 1.2.4.1.3

Factor $3$ out of $- 9 \cdot 3$ .

$y = \frac{3 \cdot 3 + 3 (- 3 \cdot 3)}{- {(3)}^{3} + 3 \cdot 3}$

Step 1.2.4.1.4

Factor $3$ out of $3 \cdot 3 + 3 (- 3 \cdot 3)$ .

$y = \frac{3 \cdot (3 - 3 \cdot 3)}{- {(3)}^{3} + 3 \cdot 3}$

Step 1.2.4.1.5

Cancel the common factors.

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Step 1.2.4.1.5.1

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

$y = \frac{3 \cdot (3 - 3 \cdot 3)}{3 (- 3^{2}) + 3 \cdot 3}$

Step 1.2.4.1.5.2

Factor $3$ out of $3 \cdot 3$ .

$y = \frac{3 \cdot (3 - 3 \cdot 3)}{3 (- 3^{2}) + 3 (3)}$

Step 1.2.4.1.5.3

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

$y = \frac{3 \cdot (3 - 3 \cdot 3)}{3 (- 3^{2} + 3)}$

Step 1.2.4.1.5.4

Cancel the common factor.

$y = \frac{3 \cdot (3 - 3 \cdot 3)}{3 (- 3^{2} + 3)}$

Step 1.2.4.1.5.5

Rewrite the expression.

$y = \frac{3 - 3 \cdot 3}{- 3^{2} + 3}$

Step 1.2.4.2

Cancel the common factor of $3 - 3 \cdot 3$ and $- 3^{2} + 3$ .

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Step 1.2.4.2.1

Factor $3$ out of $3$ .

$y = \frac{3 \cdot 1 - 3 \cdot 3}{- 3^{2} + 3}$

Step 1.2.4.2.2

Factor $3$ out of $- 3 \cdot 3$ .

$y = \frac{3 \cdot 1 + 3 (- 1 \cdot 3)}{- 3^{2} + 3}$

Step 1.2.4.2.3

Factor $3$ out of $3 (1) + 3 (- 1 \cdot 3)$ .

$y = \frac{3 (1 - 1 \cdot 3)}{- 3^{2} + 3}$

Step 1.2.4.2.4

Cancel the common factors.

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Step 1.2.4.2.4.1

Factor $3$ out of $- 3^{2}$ .

$y = \frac{3 (1 - 1 \cdot 3)}{3 (- 1 \cdot 3) + 3}$

Step 1.2.4.2.4.2

Factor $3$ out of $3$ .

$y = \frac{3 (1 - 1 \cdot 3)}{3 (- 1 \cdot 3) + 3 (1)}$

Step 1.2.4.2.4.3

Factor $3$ out of $3 (- 1 \cdot 3) + 3 (1)$ .

$y = \frac{3 (1 - 1 \cdot 3)}{3 (- 1 \cdot 3 + 1)}$

Step 1.2.4.2.4.4

Cancel the common factor.

$y = \frac{3 (1 - 1 \cdot 3)}{3 (- 1 \cdot 3 + 1)}$

Step 1.2.4.2.4.5

Rewrite the expression.

$y = \frac{1 - 1 \cdot 3}{- 1 \cdot 3 + 1}$

Step 1.2.4.3

Cancel the common factor of $1 - 1 \cdot 3$ and $- 1 \cdot 3 + 1$ .

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Step 1.2.4.3.1

Reorder terms.

$y = \frac{1 - 1 \cdot 3}{1 - 1 \cdot 3}$

Step 1.2.4.3.2

Cancel the common factor.

$y = \frac{1 - 1 \cdot 3}{1 - 1 \cdot 3}$

Step 1.2.4.3.3

Rewrite the expression.

$y = 1$

Step 2

Find the first derivative and evaluate at $x = 3$ and $y = 1$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $x^{2} - 9 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 9 x]$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $- 9$ by $1$ .

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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

Step 2.2.9

Multiply $3$ by $1$ .

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

Step 2.2.10

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

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

Step 2.2.11

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

$\frac{(3 x - x^{3}) (2 x - 9) - (x^{2} - 9 x) (3 - (3 x^{2}))}{{(3 x - x^{3})}^{2}}$

Step 2.2.12

Multiply $3$ by $- 1$ .

$\frac{(3 x - x^{3}) (2 x - 9) - (x^{2} - 9 x) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Simplify the numerator.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Expand $(3 x - x^{3}) (2 x - 9)$ using the FOIL Method.

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Step 2.3.2.1.1.1

Apply the distributive property.

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

Step 2.3.2.1.1.2

Apply the distributive property.

$\frac{3 x (2 x) + 3 x \cdot - 9 - x^{3} (2 x - 9) + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.1.3

Apply the distributive property.

$\frac{3 x (2 x) + 3 x \cdot - 9 - x^{3} (2 x) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2

Simplify each term.

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Step 2.3.2.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{3 \cdot 2 x \cdot x + 3 x \cdot - 9 - x^{3} (2 x) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.2

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

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Step 2.3.2.1.2.2.1

Move $x$ .

$\frac{3 \cdot 2 (x \cdot x) + 3 x \cdot - 9 - x^{3} (2 x) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.2.2

Multiply $x$ by $x$ .

$\frac{3 \cdot 2 x^{2} + 3 x \cdot - 9 - x^{3} (2 x) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.3

Multiply $3$ by $2$ .

$\frac{6 x^{2} + 3 x \cdot - 9 - x^{3} (2 x) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.4

Multiply $- 9$ by $3$ .

$\frac{6 x^{2} - 27 x - x^{3} (2 x) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.5

Rewrite using the commutative property of multiplication.

$\frac{6 x^{2} - 27 x - 1 \cdot 2 x^{3} x - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.6

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

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Step 2.3.2.1.2.6.1

Move $x$ .

$\frac{6 x^{2} - 27 x - 1 \cdot 2 (x \cdot x^{3}) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.6.2

Multiply $x$ by $x^{3}$ .

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Step 2.3.2.1.2.6.2.1

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

$\frac{6 x^{2} - 27 x - 1 \cdot 2 (x^{1} x^{3}) - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.6.2.2

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

$\frac{6 x^{2} - 27 x - 1 \cdot 2 x^{1 + 3} - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.6.3

Add $1$ and $3$ .

$\frac{6 x^{2} - 27 x - 1 \cdot 2 x^{4} - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.7

Multiply $- 1$ by $2$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} - x^{3} \cdot - 9 + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.2.8

Multiply $- 9$ by $- 1$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} + (- x^{2} - (- 9 x)) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.3

Multiply $- 9$ by $- 1$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} + (- x^{2} + 9 x) (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.4

Expand $(- x^{2} + 9 x) (3 - 3 x^{2})$ using the FOIL Method.

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Step 2.3.2.1.4.1

Apply the distributive property.

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - x^{2} (3 - 3 x^{2}) + 9 x (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.4.2

Apply the distributive property.

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - x^{2} \cdot 3 - x^{2} (- 3 x^{2}) + 9 x (3 - 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.4.3

Apply the distributive property.

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - x^{2} \cdot 3 - x^{2} (- 3 x^{2}) + 9 x \cdot 3 + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5

Simplify each term.

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Step 2.3.2.1.5.1

Multiply $3$ by $- 1$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} - x^{2} (- 3 x^{2}) + 9 x \cdot 3 + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.2

Rewrite using the commutative property of multiplication.

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} - 1 \cdot - 3 x^{2} x^{2} + 9 x \cdot 3 + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.3

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

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Step 2.3.2.1.5.3.1

Move $x^{2}$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} - 1 \cdot - 3 (x^{2} x^{2}) + 9 x \cdot 3 + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.3.2

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

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} - 1 \cdot - 3 x^{2 + 2} + 9 x \cdot 3 + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.3.3

Add $2$ and $2$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} - 1 \cdot - 3 x^{4} + 9 x \cdot 3 + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.4

Multiply $- 1$ by $- 3$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 9 x \cdot 3 + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.5

Multiply $3$ by $9$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x + 9 x (- 3 x^{2})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.6

Rewrite using the commutative property of multiplication.

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x + 9 \cdot - 3 x \cdot x^{2}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.7

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

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Step 2.3.2.1.5.7.1

Move $x^{2}$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x + 9 \cdot - 3 (x^{2} x)}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.7.2

Multiply $x^{2}$ by $x$ .

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Step 2.3.2.1.5.7.2.1

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

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x + 9 \cdot - 3 (x^{2} x^{1})}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.7.2.2

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

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x + 9 \cdot - 3 x^{2 + 1}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.7.3

Add $2$ and $1$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x + 9 \cdot - 3 x^{3}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.1.5.8

Multiply $9$ by $- 3$ .

$\frac{6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x - 27 x^{3}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.2

Combine the opposite terms in $6 x^{2} - 27 x - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 27 x - 27 x^{3}$ .

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Step 2.3.2.2.1

Add $- 27 x$ and $27 x$ .

$\frac{6 x^{2} - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} + 0 - 27 x^{3}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.2.2

Add $6 x^{2} - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4}$ and $0$ .

$\frac{6 x^{2} - 2 x^{4} + 9 x^{3} - 3 x^{2} + 3 x^{4} - 27 x^{3}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.3

Subtract $3 x^{2}$ from $6 x^{2}$ .

$\frac{- 2 x^{4} + 9 x^{3} + 3 x^{2} + 3 x^{4} - 27 x^{3}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.4

Add $- 2 x^{4}$ and $3 x^{4}$ .

$\frac{x^{4} + 9 x^{3} + 3 x^{2} - 27 x^{3}}{{(3 x - x^{3})}^{2}}$

Step 2.3.2.5

Subtract $27 x^{3}$ from $9 x^{3}$ .

$\frac{x^{4} - 18 x^{3} + 3 x^{2}}{{(3 x - x^{3})}^{2}}$

Step 2.3.3

Reorder terms.

$\frac{x^{4} - 18 x^{3} + 3 x^{2}}{{(- x^{3} + 3 x)}^{2}}$

Step 2.3.4

Factor $x^{2}$ out of $x^{4} - 18 x^{3} + 3 x^{2}$ .

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Step 2.3.4.1

Factor $x^{2}$ out of $x^{4}$ .

$\frac{x^{2} x^{2} - 18 x^{3} + 3 x^{2}}{{(- x^{3} + 3 x)}^{2}}$

Step 2.3.4.2

Factor $x^{2}$ out of $- 18 x^{3}$ .

$\frac{x^{2} x^{2} + x^{2} (- 18 x) + 3 x^{2}}{{(- x^{3} + 3 x)}^{2}}$

Step 2.3.4.3

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x^{2} x^{2} + x^{2} (- 18 x) + x^{2} \cdot 3}{{(- x^{3} + 3 x)}^{2}}$

Step 2.3.4.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (- 18 x)$ .

$\frac{x^{2} (x^{2} - 18 x) + x^{2} \cdot 3}{{(- x^{3} + 3 x)}^{2}}$

Step 2.3.4.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 18 x) + x^{2} \cdot 3$ .

$\frac{x^{2} (x^{2} - 18 x + 3)}{{(- x^{3} + 3 x)}^{2}}$

Step 2.3.5

Simplify the denominator.

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Step 2.3.5.1

Factor $x$ out of $- x^{3} + 3 x$ .

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Step 2.3.5.1.1

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

$\frac{x^{2} (x^{2} - 18 x + 3)}{{(x (- x^{2}) + 3 x)}^{2}}$

Step 2.3.5.1.2

Factor $x$ out of $3 x$ .

$\frac{x^{2} (x^{2} - 18 x + 3)}{{(x (- x^{2}) + x \cdot 3)}^{2}}$

Step 2.3.5.1.3

Factor $x$ out of $x (- x^{2}) + x \cdot 3$ .

$\frac{x^{2} (x^{2} - 18 x + 3)}{{(x (- x^{2} + 3))}^{2}}$

Step 2.3.5.2

Apply the product rule to $x (- x^{2} + 3)$ .

$\frac{x^{2} (x^{2} - 18 x + 3)}{x^{2} {(- x^{2} + 3)}^{2}}$

Step 2.3.6

Cancel the common factor of $x^{2}$ .

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Step 2.3.6.1

Cancel the common factor.

$\frac{x^{2} (x^{2} - 18 x + 3)}{x^{2} {(- x^{2} + 3)}^{2}}$

Step 2.3.6.2

Rewrite the expression.

$\frac{x^{2} - 18 x + 3}{{(- x^{2} + 3)}^{2}}$

Step 2.4

Evaluate the derivative at $x = 3$ .

$\frac{{(3)}^{2} - 18 \cdot 3 + 3}{{(- {(3)}^{2} + 3)}^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $3$ to the power of $2$ .

$\frac{9 - 18 \cdot 3 + 3}{{(- 3^{2} + 3)}^{2}}$

Step 2.5.1.2

Multiply $- 18$ by $3$ .

$\frac{9 - 54 + 3}{{(- 3^{2} + 3)}^{2}}$

Step 2.5.1.3

Subtract $54$ from $9$ .

$\frac{- 45 + 3}{{(- 3^{2} + 3)}^{2}}$

Step 2.5.1.4

Add $- 45$ and $3$ .

$\frac{- 42}{{(- 3^{2} + 3)}^{2}}$

Step 2.5.2

Simplify the denominator.

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Step 2.5.2.1

Raise $3$ to the power of $2$ .

$\frac{- 42}{{(- 1 \cdot 9 + 3)}^{2}}$

Step 2.5.2.2

Multiply $- 1$ by $9$ .

$\frac{- 42}{{(- 9 + 3)}^{2}}$

Step 2.5.2.3

Add $- 9$ and $3$ .

$\frac{- 42}{{(- 6)}^{2}}$

Step 2.5.2.4

Raise $- 6$ to the power of $2$ .

$\frac{- 42}{36}$

Step 2.5.3

Reduce the expression by cancelling the common factors.

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Step 2.5.3.1

Cancel the common factor of $- 42$ and $36$ .

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Step 2.5.3.1.1

Factor $6$ out of $- 42$ .

$\frac{6 (- 7)}{36}$

Step 2.5.3.1.2

Cancel the common factors.

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Step 2.5.3.1.2.1

Factor $6$ out of $36$ .

$\frac{6 \cdot - 7}{6 \cdot 6}$

Step 2.5.3.1.2.2

Cancel the common factor.

$\frac{6 \cdot - 7}{6 \cdot 6}$

Step 2.5.3.1.2.3

Rewrite the expression.

$\frac{- 7}{6}$

Step 2.5.3.2

Move the negative in front of the fraction.

$- \frac{7}{6}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $- \frac{7}{6}$ and a given point $(3, 1)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{7}{6} \cdot (x - (3))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 1 = - \frac{7}{6} \cdot (x - 3)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{7}{6} \cdot (x - 3)$ .

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Step 3.3.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{7}{6} \cdot (x - 3)$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$y - 1 = - \frac{7}{6} x - \frac{7}{6} \cdot - 3$

Step 3.3.1.2.2

Combine $x$ and $\frac{7}{6}$ .

$y - 1 = - \frac{x \cdot 7}{6} - \frac{7}{6} \cdot - 3$

Step 3.3.1.2.3

Cancel the common factor of $3$ .

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Step 3.3.1.2.3.1

Move the leading negative in $- \frac{7}{6}$ into the numerator.

$y - 1 = - \frac{x \cdot 7}{6} + \frac{- 7}{6} \cdot - 3$

Step 3.3.1.2.3.2

Factor $3$ out of $6$ .

$y - 1 = - \frac{x \cdot 7}{6} + \frac{- 7}{3 (2)} \cdot - 3$

Step 3.3.1.2.3.3

Factor $3$ out of $- 3$ .

$y - 1 = - \frac{x \cdot 7}{6} + \frac{- 7}{3 \cdot 2} \cdot (3 \cdot - 1)$

Step 3.3.1.2.3.4

Cancel the common factor.

$y - 1 = - \frac{x \cdot 7}{6} + \frac{- 7}{3 \cdot 2} \cdot (3 \cdot - 1)$

Step 3.3.1.2.3.5

Rewrite the expression.

$y - 1 = - \frac{x \cdot 7}{6} + \frac{- 7}{2} \cdot - 1$

Step 3.3.1.2.4

Combine $\frac{- 7}{2}$ and $- 1$ .

$y - 1 = - \frac{x \cdot 7}{6} + \frac{- 7 \cdot - 1}{2}$

Step 3.3.1.2.5

Multiply $- 7$ by $- 1$ .

$y - 1 = - \frac{x \cdot 7}{6} + \frac{7}{2}$

Step 3.3.1.3

Move $7$ to the left of $x$ .

$y - 1 = - \frac{7 x}{6} + \frac{7}{2}$

Step 3.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.2.1

Add $1$ to both sides of the equation.

$y = - \frac{7 x}{6} + \frac{7}{2} + 1$

Step 3.3.2.2

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

$y = - \frac{7 x}{6} + \frac{7}{2} + \frac{2}{2}$

Step 3.3.2.3

Combine the numerators over the common denominator.

$y = - \frac{7 x}{6} + \frac{7 + 2}{2}$

Step 3.3.2.4

Add $7$ and $2$ .

$y = - \frac{7 x}{6} + \frac{9}{2}$

Step 3.3.3

Write in $y = m x + b$ form.

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{7}{6} x) + \frac{9}{2}$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{7}{6} x + \frac{9}{2}$

Step 4