Calculus Examples

$y = \frac{x^{2} - 16 x}{4 x - x^{3}}$ at $x = 4$

Step 1

Find the corresponding $y$ -value to $x = 4$ .

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Step 1.1

Substitute $4$ in for $x$ .

$y = \frac{{(4)}^{2} - 16 \cdot 4}{4 (4) - {(4)}^{3}}$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = \frac{4^{2} - 16 \cdot 4}{4 (4) - {(4)}^{3}}$

Step 1.2.2

Remove parentheses.

$y = \frac{4^{2} - 16 \cdot 4}{4 (4) - 4^{3}}$

Step 1.2.3

Remove parentheses.

$y = \frac{{(4)}^{2} - 16 \cdot 4}{4 (4) - {(4)}^{3}}$

Step 1.2.4

Simplify $\frac{{(4)}^{2} - 16 \cdot 4}{4 (4) - {(4)}^{3}}$ .

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Step 1.2.4.1

Cancel the common factor of ${(4)}^{2} - 16 \cdot 4$ and $4 (4) - {(4)}^{3}$ .

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Step 1.2.4.1.1

Reorder terms.

$y = \frac{{(4)}^{2} - 16 \cdot 4}{- {(4)}^{3} + 4 \cdot 4}$

Step 1.2.4.1.2

Factor $4$ out of ${(4)}^{2}$ .

$y = \frac{4 \cdot 4 - 16 \cdot 4}{- {(4)}^{3} + 4 \cdot 4}$

Step 1.2.4.1.3

Factor $4$ out of $- 16 \cdot 4$ .

$y = \frac{4 \cdot 4 + 4 (- 4 \cdot 4)}{- {(4)}^{3} + 4 \cdot 4}$

Step 1.2.4.1.4

Factor $4$ out of $4 \cdot 4 + 4 (- 4 \cdot 4)$ .

$y = \frac{4 \cdot (4 - 4 \cdot 4)}{- {(4)}^{3} + 4 \cdot 4}$

Step 1.2.4.1.5

Cancel the common factors.

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Step 1.2.4.1.5.1

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

$y = \frac{4 \cdot (4 - 4 \cdot 4)}{4 (- 4^{2}) + 4 \cdot 4}$

Step 1.2.4.1.5.2

Factor $4$ out of $4 \cdot 4$ .

$y = \frac{4 \cdot (4 - 4 \cdot 4)}{4 (- 4^{2}) + 4 (4)}$

Step 1.2.4.1.5.3

Factor $4$ out of $4 (- 4^{2}) + 4 (4)$ .

$y = \frac{4 \cdot (4 - 4 \cdot 4)}{4 (- 4^{2} + 4)}$

Step 1.2.4.1.5.4

Cancel the common factor.

$y = \frac{4 \cdot (4 - 4 \cdot 4)}{4 (- 4^{2} + 4)}$

Step 1.2.4.1.5.5

Rewrite the expression.

$y = \frac{4 - 4 \cdot 4}{- 4^{2} + 4}$

Step 1.2.4.2

Cancel the common factor of $4 - 4 \cdot 4$ and $- 4^{2} + 4$ .

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Step 1.2.4.2.1

Factor $4$ out of $4$ .

$y = \frac{4 \cdot 1 - 4 \cdot 4}{- 4^{2} + 4}$

Step 1.2.4.2.2

Factor $4$ out of $- 4 \cdot 4$ .

$y = \frac{4 \cdot 1 + 4 (- 1 \cdot 4)}{- 4^{2} + 4}$

Step 1.2.4.2.3

Factor $4$ out of $4 (1) + 4 (- 1 \cdot 4)$ .

$y = \frac{4 (1 - 1 \cdot 4)}{- 4^{2} + 4}$

Step 1.2.4.2.4

Cancel the common factors.

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Step 1.2.4.2.4.1

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

$y = \frac{4 (1 - 1 \cdot 4)}{4 (- 1 \cdot 4) + 4}$

Step 1.2.4.2.4.2

Factor $4$ out of $4$ .

$y = \frac{4 (1 - 1 \cdot 4)}{4 (- 1 \cdot 4) + 4 (1)}$

Step 1.2.4.2.4.3

Factor $4$ out of $4 (- 1 \cdot 4) + 4 (1)$ .

$y = \frac{4 (1 - 1 \cdot 4)}{4 (- 1 \cdot 4 + 1)}$

Step 1.2.4.2.4.4

Cancel the common factor.

$y = \frac{4 (1 - 1 \cdot 4)}{4 (- 1 \cdot 4 + 1)}$

Step 1.2.4.2.4.5

Rewrite the expression.

$y = \frac{1 - 1 \cdot 4}{- 1 \cdot 4 + 1}$

Step 1.2.4.3

Cancel the common factor of $1 - 1 \cdot 4$ and $- 1 \cdot 4 + 1$ .

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Step 1.2.4.3.1

Reorder terms.

$y = \frac{1 - 1 \cdot 4}{1 - 1 \cdot 4}$

Step 1.2.4.3.2

Cancel the common factor.

$y = \frac{1 - 1 \cdot 4}{1 - 1 \cdot 4}$

Step 1.2.4.3.3

Rewrite the expression.

$y = 1$

Step 2

Find the first derivative and evaluate at $x = 4$ 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 $f (x) = x^{2} - 16 x$ and $g (x) = 4 x - x^{3}$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $- 16$ by $1$ .

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

Step 2.2.6

By the Sum Rule, the derivative of $4 x - x^{3}$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- x^{3}]$ .

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

Step 2.2.7

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 2.2.8

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

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

Step 2.2.9

Multiply $4$ by $1$ .

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

Step 2.2.10

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

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

Step 2.2.11

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

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

Step 2.2.12

Multiply $3$ by $- 1$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$\frac{(4 x - x^{3}) (2 x - 16) + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

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 $(4 x - x^{3}) (2 x - 16)$ using the FOIL Method.

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Step 2.3.2.1.1.1

Apply the distributive property.

$\frac{4 x (2 x - 16) - x^{3} (2 x - 16) + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.1.2

Apply the distributive property.

$\frac{4 x (2 x) + 4 x \cdot - 16 - x^{3} (2 x - 16) + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.1.3

Apply the distributive property.

$\frac{4 x (2 x) + 4 x \cdot - 16 - x^{3} (2 x) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

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.

$\frac{4 \cdot 2 x \cdot x + 4 x \cdot - 16 - x^{3} (2 x) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

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 $x$ .

$\frac{4 \cdot 2 (x \cdot x) + 4 x \cdot - 16 - x^{3} (2 x) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.2.2

Multiply $x$ by $x$ .

$\frac{4 \cdot 2 x^{2} + 4 x \cdot - 16 - x^{3} (2 x) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.3

Multiply $4$ by $2$ .

$\frac{8 x^{2} + 4 x \cdot - 16 - x^{3} (2 x) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.4

Multiply $- 16$ by $4$ .

$\frac{8 x^{2} - 64 x - x^{3} (2 x) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.5

Rewrite using the commutative property of multiplication.

$\frac{8 x^{2} - 64 x - 1 \cdot 2 x^{3} x - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

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 $x$ .

$\frac{8 x^{2} - 64 x - 1 \cdot 2 (x \cdot x^{3}) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

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 $1$ .

$\frac{8 x^{2} - 64 x - 1 \cdot 2 (x^{1} x^{3}) - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.6.2.2

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

$\frac{8 x^{2} - 64 x - 1 \cdot 2 x^{1 + 3} - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.6.3

Add $1$ and $3$ .

$\frac{8 x^{2} - 64 x - 1 \cdot 2 x^{4} - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.7

Multiply $- 1$ by $2$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} - x^{3} \cdot - 16 + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.2.8

Multiply $- 16$ by $- 1$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} + (- x^{2} - (- 16 x)) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.3

Multiply $- 16$ by $- 1$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} + (- x^{2} + 16 x) (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.4

Expand $(- x^{2} + 16 x) (4 - 3 x^{2})$ using the FOIL Method.

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Step 2.3.2.1.4.1

Apply the distributive property.

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - x^{2} (4 - 3 x^{2}) + 16 x (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.4.2

Apply the distributive property.

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - x^{2} \cdot 4 - x^{2} (- 3 x^{2}) + 16 x (4 - 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.4.3

Apply the distributive property.

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - x^{2} \cdot 4 - x^{2} (- 3 x^{2}) + 16 x \cdot 4 + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5

Simplify each term.

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Step 2.3.2.1.5.1

Multiply $4$ by $- 1$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} - x^{2} (- 3 x^{2}) + 16 x \cdot 4 + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.2

Rewrite using the commutative property of multiplication.

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} - 1 \cdot - 3 x^{2} x^{2} + 16 x \cdot 4 + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

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 $x^{2}$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} - 1 \cdot - 3 (x^{2} x^{2}) + 16 x \cdot 4 + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.3.2

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

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} - 1 \cdot - 3 x^{2 + 2} + 16 x \cdot 4 + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.3.3

Add $2$ and $2$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} - 1 \cdot - 3 x^{4} + 16 x \cdot 4 + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.4

Multiply $- 1$ by $- 3$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 16 x \cdot 4 + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.5

Multiply $4$ by $16$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x + 16 x (- 3 x^{2})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.6

Rewrite using the commutative property of multiplication.

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x + 16 \cdot - 3 x \cdot x^{2}}{{(4 x - x^{3})}^{2}}$

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 $x^{2}$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x + 16 \cdot - 3 (x^{2} x)}{{(4 x - x^{3})}^{2}}$

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 $1$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x + 16 \cdot - 3 (x^{2} x^{1})}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.7.2.2

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

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x + 16 \cdot - 3 x^{2 + 1}}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.7.3

Add $2$ and $1$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x + 16 \cdot - 3 x^{3}}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.1.5.8

Multiply $16$ by $- 3$ .

$\frac{8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x - 48 x^{3}}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.2

Combine the opposite terms in $8 x^{2} - 64 x - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 64 x - 48 x^{3}$ .

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Step 2.3.2.2.1

Add $- 64 x$ and $64 x$ .

$\frac{8 x^{2} - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} + 0 - 48 x^{3}}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.2.2

Add $8 x^{2} - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4}$ and $0$ .

$\frac{8 x^{2} - 2 x^{4} + 16 x^{3} - 4 x^{2} + 3 x^{4} - 48 x^{3}}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.3

Subtract $4 x^{2}$ from $8 x^{2}$ .

$\frac{- 2 x^{4} + 16 x^{3} + 4 x^{2} + 3 x^{4} - 48 x^{3}}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.4

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

$\frac{x^{4} + 16 x^{3} + 4 x^{2} - 48 x^{3}}{{(4 x - x^{3})}^{2}}$

Step 2.3.2.5

Subtract $48 x^{3}$ from $16 x^{3}$ .

$\frac{x^{4} - 32 x^{3} + 4 x^{2}}{{(4 x - x^{3})}^{2}}$

Step 2.3.3

Reorder terms.

$\frac{x^{4} - 32 x^{3} + 4 x^{2}}{{(- x^{3} + 4 x)}^{2}}$

Step 2.3.4

Factor $x^{2}$ out of $x^{4} - 32 x^{3} + 4 x^{2}$ .

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Step 2.3.4.1

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

$\frac{x^{2} x^{2} - 32 x^{3} + 4 x^{2}}{{(- x^{3} + 4 x)}^{2}}$

Step 2.3.4.2

Factor $x^{2}$ out of $- 32 x^{3}$ .

$\frac{x^{2} x^{2} + x^{2} (- 32 x) + 4 x^{2}}{{(- x^{3} + 4 x)}^{2}}$

Step 2.3.4.3

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

$\frac{x^{2} x^{2} + x^{2} (- 32 x) + x^{2} \cdot 4}{{(- x^{3} + 4 x)}^{2}}$

Step 2.3.4.4

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} (- 32 x)$ .

$\frac{x^{2} (x^{2} - 32 x) + x^{2} \cdot 4}{{(- x^{3} + 4 x)}^{2}}$

Step 2.3.4.5

Factor $x^{2}$ out of $x^{2} (x^{2} - 32 x) + x^{2} \cdot 4$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(- x^{3} + 4 x)}^{2}}$

Step 2.3.5

Simplify the denominator.

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Step 2.3.5.1

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

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Step 2.3.5.1.1

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x (- x^{2}) + 4 x)}^{2}}$

Step 2.3.5.1.2

Factor $x$ out of $4 x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x (- x^{2}) + x \cdot 4)}^{2}}$

Step 2.3.5.1.3

Factor $x$ out of $x (- x^{2}) + x \cdot 4$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x (- x^{2} + 4))}^{2}}$

Step 2.3.5.2

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x (- x^{2} + 2^{2}))}^{2}}$

Step 2.3.5.3

Reorder $- x^{2}$ and $2^{2}$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x (2^{2} - x^{2}))}^{2}}$

Step 2.3.5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x (2 + x) (2 - x))}^{2}}$

Step 2.3.5.5

Apply the product rule to $x (2 + x) (2 - x)$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x (2 + x))}^{2} {(2 - x)}^{2}}$

Step 2.3.5.6

Apply the distributive property.

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(x \cdot 2 + x \cdot x)}^{2} {(2 - x)}^{2}}$

Step 2.3.5.7

Move $2$ to the left of $x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(2 \cdot x + x \cdot x)}^{2} {(2 - x)}^{2}}$

Step 2.3.5.8

Multiply $x$ by $x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{{(2 \cdot x + x^{2})}^{2} {(2 - x)}^{2}}$

Step 2.3.5.9

Rewrite ${(2 x + x^{2})}^{2}$ as $(2 x + x^{2}) (2 x + x^{2})$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(2 x + x^{2}) (2 x + x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.10

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

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Step 2.3.5.10.1

Apply the distributive property.

$\frac{x^{2} (x^{2} - 32 x + 4)}{(2 x (2 x + x^{2}) + x^{2} (2 x + x^{2})) {(2 - x)}^{2}}$

Step 2.3.5.10.2

Apply the distributive property.

$\frac{x^{2} (x^{2} - 32 x + 4)}{(2 x (2 x) + 2 x \cdot x^{2} + x^{2} (2 x + x^{2})) {(2 - x)}^{2}}$

Step 2.3.5.10.3

Apply the distributive property.

$\frac{x^{2} (x^{2} - 32 x + 4)}{(2 x (2 x) + 2 x \cdot x^{2} + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11

Simplify and combine like terms.

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Step 2.3.5.11.1

Simplify each term.

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Step 2.3.5.11.1.1

Rewrite using the commutative property of multiplication.

$\frac{x^{2} (x^{2} - 32 x + 4)}{(2 \cdot 2 x \cdot x + 2 x \cdot x^{2} + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.2

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

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Step 2.3.5.11.1.2.1

Move $x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(2 \cdot 2 (x \cdot x) + 2 x \cdot x^{2} + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.2.2

Multiply $x$ by $x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(2 \cdot 2 x^{2} + 2 x \cdot x^{2} + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.3

Multiply $2$ by $2$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x \cdot x^{2} + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.4

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

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Step 2.3.5.11.1.4.1

Move $x^{2}$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 (x^{2} x) + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.4.2

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

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Step 2.3.5.11.1.4.2.1

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 (x^{2} x^{1}) + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.4.2.2

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{2 + 1} + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.4.3

Add $2$ and $1$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + x^{2} (2 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.5

Rewrite using the commutative property of multiplication.

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + 2 x^{2} x + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.6

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

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Step 2.3.5.11.1.6.1

Move $x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + 2 (x \cdot x^{2}) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.6.2

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

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Step 2.3.5.11.1.6.2.1

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + 2 (x^{1} x^{2}) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.6.2.2

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + 2 x^{1 + 2} + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.6.3

Add $1$ and $2$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + 2 x^{3} + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.7

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

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Step 2.3.5.11.1.7.1

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + 2 x^{3} + x^{2 + 2}) {(2 - x)}^{2}}$

Step 2.3.5.11.1.7.2

Add $2$ and $2$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 2 x^{3} + 2 x^{3} + x^{4}) {(2 - x)}^{2}}$

Step 2.3.5.11.2

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(4 x^{2} + 4 x^{3} + x^{4}) {(2 - x)}^{2}}$

Step 2.3.5.12

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

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Step 2.3.5.12.1

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(x^{2} \cdot 4 + 4 x^{3} + x^{4}) {(2 - x)}^{2}}$

Step 2.3.5.12.2

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(x^{2} \cdot 4 + x^{2} (4 x) + x^{4}) {(2 - x)}^{2}}$

Step 2.3.5.12.3

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

$\frac{x^{2} (x^{2} - 32 x + 4)}{(x^{2} \cdot 4 + x^{2} (4 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.12.4

Factor $x^{2}$ out of $x^{2} \cdot 4 + x^{2} (4 x)$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{(x^{2} \cdot (4 + 4 x) + x^{2} x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.12.5

Factor $x^{2}$ out of $x^{2} \cdot (4 + 4 x) + x^{2} x^{2}$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{x^{2} (4 + 4 x + x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.13

Factor using the perfect square rule.

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Step 2.3.5.13.1

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{x^{2} (2^{2} + 4 x + x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.13.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot 2 \cdot x$

Step 2.3.5.13.3

Rewrite the polynomial.

$\frac{x^{2} (x^{2} - 32 x + 4)}{x^{2} (2^{2} + 2 \cdot 2 \cdot x + x^{2}) {(2 - x)}^{2}}$

Step 2.3.5.13.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2$ and $b = x$ .

$\frac{x^{2} (x^{2} - 32 x + 4)}{x^{2} {(2 + x)}^{2} {(2 - x)}^{2}}$

Step 2.3.6

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

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Step 2.3.6.1

Cancel the common factor.

$\frac{x^{2} (x^{2} - 32 x + 4)}{x^{2} {(2 + x)}^{2} {(2 - x)}^{2}}$

Step 2.3.6.2

Rewrite the expression.

$\frac{x^{2} - 32 x + 4}{{(2 + x)}^{2} {(2 - x)}^{2}}$

Step 2.4

Evaluate the derivative at $x = 4$ .

$\frac{{(4)}^{2} - 32 \cdot 4 + 4}{{(2 + 4)}^{2} {(2 - (4))}^{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Remove parentheses.

$\frac{{(4)}^{2} - 32 \cdot 4 + 4}{{(2 + 4)}^{2} {(2 - (4))}^{2}}$

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

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

$\frac{16 - 32 \cdot 4 + 4}{{(2 + 4)}^{2} {(2 - (4))}^{2}}$

Step 2.5.2.2

Multiply $- 32$ by $4$ .

$\frac{16 - 128 + 4}{{(2 + 4)}^{2} {(2 - (4))}^{2}}$

Step 2.5.2.3

Subtract $128$ from $16$ .

$\frac{- 112 + 4}{{(2 + 4)}^{2} {(2 - (4))}^{2}}$

Step 2.5.2.4

Add $- 112$ and $4$ .

$\frac{- 108}{{(2 + 4)}^{2} {(2 - (4))}^{2}}$

Step 2.5.3

Simplify the denominator.

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Step 2.5.3.1

Add $2$ and $4$ .

$\frac{- 108}{6^{2} {(2 - (4))}^{2}}$

Step 2.5.3.2

Multiply $- 1$ by $4$ .

$\frac{- 108}{6^{2} {(2 - 4)}^{2}}$

Step 2.5.3.3

Subtract $4$ from $2$ .

$\frac{- 108}{6^{2} {(- 2)}^{2}}$

Step 2.5.3.4

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

$\frac{- 108}{36 {(- 2)}^{2}}$

Step 2.5.3.5

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

$\frac{- 108}{36 \cdot 4}$

Step 2.5.4

Reduce the expression by cancelling the common factors.

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Step 2.5.4.1

Multiply $36$ by $4$ .

$\frac{- 108}{144}$

Step 2.5.4.2

Cancel the common factor of $- 108$ and $144$ .

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Step 2.5.4.2.1

Factor $36$ out of $- 108$ .

$\frac{36 (- 3)}{144}$

Step 2.5.4.2.2

Cancel the common factors.

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Step 2.5.4.2.2.1

Factor $36$ out of $144$ .

$\frac{36 \cdot - 3}{36 \cdot 4}$

Step 2.5.4.2.2.2

Cancel the common factor.

$\frac{36 \cdot - 3}{36 \cdot 4}$

Step 2.5.4.2.2.3

Rewrite the expression.

$\frac{- 3}{4}$

Step 2.5.4.3

Move the negative in front of the fraction.

$- \frac{3}{4}$

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 $- \frac{3}{4}$ and a given point $(4, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{3}{4} \cdot (x - (4))$

Step 3.2

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

$y - 1 = - \frac{3}{4} \cdot (x - 4)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- \frac{3}{4} \cdot (x - 4)$ .

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Step 3.3.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{3}{4} \cdot (x - 4)$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$y - 1 = - \frac{3}{4} x - \frac{3}{4} \cdot - 4$

Step 3.3.1.2.2

Combine $x$ and $\frac{3}{4}$ .

$y - 1 = - \frac{x \cdot 3}{4} - \frac{3}{4} \cdot - 4$

Step 3.3.1.2.3

Cancel the common factor of $4$ .

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Step 3.3.1.2.3.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$y - 1 = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot - 4$

Step 3.3.1.2.3.2

Factor $4$ out of $- 4$ .

$y - 1 = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot (4 (- 1))$

Step 3.3.1.2.3.3

Cancel the common factor.

$y - 1 = - \frac{x \cdot 3}{4} + \frac{- 3}{4} \cdot (4 \cdot - 1)$

Step 3.3.1.2.3.4

Rewrite the expression.

$y - 1 = - \frac{x \cdot 3}{4} - 3 \cdot - 1$

Step 3.3.1.2.4

Multiply $- 3$ by $- 1$ .

$y - 1 = - \frac{x \cdot 3}{4} + 3$

Step 3.3.1.3

Move $3$ to the left of $x$ .

$y - 1 = - \frac{3 x}{4} + 3$

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.

$y = - \frac{3 x}{4} + 3 + 1$

Step 3.3.2.2

Add $3$ and $1$ .

$y = - \frac{3 x}{4} + 4$

Step 3.3.3

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

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Step 3.3.3.1

Reorder terms.

$y = - (\frac{3}{4} x) + 4$

Step 3.3.3.2

Remove parentheses.

$y = - \frac{3}{4} x + 4$

Step 4