Calculus Examples

$y = \frac{(x + 1) (x - 8)}{(x - 1) (x + 8)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{(x + 1) (x - 8)}{(x - 1) (x + 8)})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = (x + 1) (x - 8)$ and $g (x) = (x - 1) (x + 8)$ .

$\frac{(x - 1) (x + 8) \frac{d}{d x} [(x + 1) (x - 8)] - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.2

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 $f (x) = x + 1$ and $g (x) = x - 8$ .

$\frac{(x - 1) (x + 8) ((x + 1) \frac{d}{d x} [x - 8] + (x - 8) \frac{d}{d x} [x + 1]) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $x - 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$ .

$\frac{(x - 1) (x + 8) ((x + 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 8]) + (x - 8) \frac{d}{d x} [x + 1]) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.2

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

$\frac{(x - 1) (x + 8) ((x + 1) (1 + \frac{d}{d x} [- 8]) + (x - 8) \frac{d}{d x} [x + 1]) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{(x - 1) (x + 8) ((x + 1) (1 + 0) + (x - 8) \frac{d}{d x} [x + 1]) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Add $1$ and $0$ .

$\frac{(x - 1) (x + 8) ((x + 1) \cdot 1 + (x - 8) \frac{d}{d x} [x + 1]) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.4.2

Multiply $x + 1$ by $1$ .

$\frac{(x - 1) (x + 8) (x + 1 + (x - 8) \frac{d}{d x} [x + 1]) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.5

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

$\frac{(x - 1) (x + 8) (x + 1 + (x - 8) (\frac{d}{d x} [x] + \frac{d}{d x} [1])) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.6

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

$\frac{(x - 1) (x + 8) (x + 1 + (x - 8) (1 + \frac{d}{d x} [1])) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.7

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

$\frac{(x - 1) (x + 8) (x + 1 + (x - 8) (1 + 0)) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.8

Simplify by adding terms.

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Step 3.3.8.1

Add $1$ and $0$ .

$\frac{(x - 1) (x + 8) (x + 1 + (x - 8) \cdot 1) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.8.2

Multiply $x - 8$ by $1$ .

$\frac{(x - 1) (x + 8) (x + 1 + x - 8) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.8.3

Add $x$ and $x$ .

$\frac{(x - 1) (x + 8) (2 x + 1 - 8) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.3.8.4

Subtract $8$ from $1$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) \frac{d}{d x} [(x - 1) (x + 8)]}{{((x - 1) (x + 8))}^{2}}$

Step 3.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = x + 8$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) ((x - 1) \frac{d}{d x} [x + 8] + (x + 8) \frac{d}{d x} [x - 1])}{{((x - 1) (x + 8))}^{2}}$

Step 3.5

Differentiate.

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Step 3.5.1

By the Sum Rule, the derivative of $x + 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [8]$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) ((x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [8]) + (x + 8) \frac{d}{d x} [x - 1])}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.2

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

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) ((x - 1) (1 + \frac{d}{d x} [8]) + (x + 8) \frac{d}{d x} [x - 1])}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.3

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) ((x - 1) (1 + 0) + (x + 8) \frac{d}{d x} [x - 1])}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.4

Simplify the expression.

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Step 3.5.4.1

Add $1$ and $0$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) ((x - 1) \cdot 1 + (x + 8) \frac{d}{d x} [x - 1])}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.4.2

Multiply $x - 1$ by $1$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (x - 1 + (x + 8) \frac{d}{d x} [x - 1])}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.5

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

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (x - 1 + (x + 8) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]))}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.6

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

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (x - 1 + (x + 8) (1 + \frac{d}{d x} [- 1]))}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.7

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

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (x - 1 + (x + 8) (1 + 0))}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.8

Simplify by adding terms.

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Step 3.5.8.1

Add $1$ and $0$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (x - 1 + (x + 8) \cdot 1)}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.8.2

Multiply $x + 8$ by $1$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (x - 1 + x + 8)}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.8.3

Add $x$ and $x$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (2 x - 1 + 8)}{{((x - 1) (x + 8))}^{2}}$

Step 3.5.8.4

Add $- 1$ and $8$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (2 x + 7)}{{((x - 1) (x + 8))}^{2}}$

Step 3.6

Simplify.

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Step 3.6.1

Apply the product rule to $(x - 1) (x + 8)$ .

$\frac{(x - 1) (x + 8) (2 x - 7) - (x + 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.2

Apply the distributive property.

$\frac{(x - 1) (x + 8) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3

Simplify the numerator.

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Step 3.6.3.1

Simplify each term.

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Step 3.6.3.1.1

Expand $(x - 1) (x + 8)$ using the FOIL Method.

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Step 3.6.3.1.1.1

Apply the distributive property.

$\frac{(x (x + 8) - 1 (x + 8)) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.1.2

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 8 - 1 (x + 8)) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.1.3

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 8 - 1 x - 1 \cdot 8) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.2

Simplify and combine like terms.

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Step 3.6.3.1.2.1

Simplify each term.

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Step 3.6.3.1.2.1.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + x \cdot 8 - 1 x - 1 \cdot 8) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.2.1.2

Move $8$ to the left of $x$ .

$\frac{(x^{2} + 8 \cdot x - 1 x - 1 \cdot 8) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.2.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{(x^{2} + 8 x - x - 1 \cdot 8) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.2.1.4

Multiply $- 1$ by $8$ .

$\frac{(x^{2} + 8 x - x - 8) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.2.2

Subtract $x$ from $8 x$ .

$\frac{(x^{2} + 7 x - 8) (2 x - 7) + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.3

Expand $(x^{2} + 7 x - 8) (2 x - 7)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{x^{2} (2 x) + x^{2} \cdot - 7 + 7 x (2 x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4

Simplify each term.

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Step 3.6.3.1.4.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{2} x + x^{2} \cdot - 7 + 7 x (2 x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.2

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

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Step 3.6.3.1.4.2.1

Move $x$ .

$\frac{2 (x \cdot x^{2}) + x^{2} \cdot - 7 + 7 x (2 x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.2.2

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

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Step 3.6.3.1.4.2.2.1

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

$\frac{2 (x^{1} x^{2}) + x^{2} \cdot - 7 + 7 x (2 x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.2.2.2

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

$\frac{2 x^{1 + 2} + x^{2} \cdot - 7 + 7 x (2 x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + x^{2} \cdot - 7 + 7 x (2 x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.3

Move $- 7$ to the left of $x^{2}$ .

$\frac{2 x^{3} - 7 \cdot x^{2} + 7 x (2 x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.4

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 7 x^{2} + 7 \cdot 2 x \cdot x + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.5

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

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Step 3.6.3.1.4.5.1

Move $x$ .

$\frac{2 x^{3} - 7 x^{2} + 7 \cdot 2 (x \cdot x) + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.5.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 7 x^{2} + 7 \cdot 2 x^{2} + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.6

Multiply $7$ by $2$ .

$\frac{2 x^{3} - 7 x^{2} + 14 x^{2} + 7 x \cdot - 7 - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.7

Multiply $- 7$ by $7$ .

$\frac{2 x^{3} - 7 x^{2} + 14 x^{2} - 49 x - 8 (2 x) - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.8

Multiply $2$ by $- 8$ .

$\frac{2 x^{3} - 7 x^{2} + 14 x^{2} - 49 x - 16 x - 8 \cdot - 7 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.4.9

Multiply $- 8$ by $- 7$ .

$\frac{2 x^{3} - 7 x^{2} + 14 x^{2} - 49 x - 16 x + 56 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.5

Add $- 7 x^{2}$ and $14 x^{2}$ .

$\frac{2 x^{3} + 7 x^{2} - 49 x - 16 x + 56 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.6

Subtract $16 x$ from $- 49 x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x - 1 \cdot 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.7

Multiply $- 1$ by $1$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x - 1) (x - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.8

Expand $(- x - 1) (x - 8)$ using the FOIL Method.

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Step 3.6.3.1.8.1

Apply the distributive property.

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x (x - 8) - 1 (x - 8)) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.8.2

Apply the distributive property.

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x \cdot x - x \cdot - 8 - 1 (x - 8)) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.8.3

Apply the distributive property.

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x \cdot x - x \cdot - 8 - 1 x - 1 \cdot - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.9

Simplify and combine like terms.

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Step 3.6.3.1.9.1

Simplify each term.

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Step 3.6.3.1.9.1.1

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

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Step 3.6.3.1.9.1.1.1

Move $x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- (x \cdot x) - x \cdot - 8 - 1 x - 1 \cdot - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.9.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x^{2} - x \cdot - 8 - 1 x - 1 \cdot - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.9.1.2

Multiply $- 8$ by $- 1$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x^{2} + 8 x - 1 x - 1 \cdot - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.9.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x^{2} + 8 x - x - 1 \cdot - 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.9.1.4

Multiply $- 1$ by $- 8$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x^{2} + 8 x - x + 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.9.2

Subtract $x$ from $8 x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 + (- x^{2} + 7 x + 8) (2 x + 7)}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.10

Expand $(- x^{2} + 7 x + 8) (2 x + 7)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - x^{2} (2 x) - x^{2} \cdot 7 + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11

Simplify each term.

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Step 3.6.3.1.11.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 1 \cdot 2 x^{2} x - x^{2} \cdot 7 + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.2

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

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Step 3.6.3.1.11.2.1

Move $x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot 7 + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.2.2

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

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Step 3.6.3.1.11.2.2.1

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

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot 7 + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.2.2.2

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

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot 7 + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 1 \cdot 2 x^{3} - x^{2} \cdot 7 + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - x^{2} \cdot 7 + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.4

Multiply $7$ by $- 1$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 7 x (2 x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 7 \cdot 2 x \cdot x + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.6

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

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Step 3.6.3.1.11.6.1

Move $x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 7 \cdot 2 (x \cdot x) + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 7 \cdot 2 x^{2} + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.7

Multiply $7$ by $2$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 14 x^{2} + 7 x \cdot 7 + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.8

Multiply $7$ by $7$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 14 x^{2} + 49 x + 8 (2 x) + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.9

Multiply $2$ by $8$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 14 x^{2} + 49 x + 16 x + 8 \cdot 7}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.11.10

Multiply $8$ by $7$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} - 7 x^{2} + 14 x^{2} + 49 x + 16 x + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.12

Add $- 7 x^{2}$ and $14 x^{2}$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} + 7 x^{2} + 49 x + 16 x + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.1.13

Add $49 x$ and $16 x$ .

$\frac{2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} + 7 x^{2} + 65 x + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.2

Combine the opposite terms in $2 x^{3} + 7 x^{2} - 65 x + 56 - 2 x^{3} + 7 x^{2} + 65 x + 56$ .

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Step 3.6.3.2.1

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

$\frac{7 x^{2} - 65 x + 56 + 0 + 7 x^{2} + 65 x + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.2.2

Add $7 x^{2} - 65 x + 56$ and $0$ .

$\frac{7 x^{2} - 65 x + 56 + 7 x^{2} + 65 x + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.2.3

Add $- 65 x$ and $65 x$ .

$\frac{7 x^{2} + 56 + 7 x^{2} + 0 + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.2.4

Add $7 x^{2} + 56 + 7 x^{2}$ and $0$ .

$\frac{7 x^{2} + 56 + 7 x^{2} + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.3

Add $7 x^{2}$ and $7 x^{2}$ .

$\frac{14 x^{2} + 56 + 56}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.3.4

Add $56$ and $56$ .

$\frac{14 x^{2} + 112}{{(x - 1)}^{2} {(x + 8)}^{2}}$

Step 3.6.4

Reorder terms.

$\frac{14 x^{2} + 112}{{(x + 8)}^{2} {(x - 1)}^{2}}$

Step 3.6.5

Factor $14$ out of $14 x^{2} + 112$ .

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Step 3.6.5.1

Factor $14$ out of $14 x^{2}$ .

$\frac{14 (x^{2}) + 112}{{(x + 8)}^{2} {(x - 1)}^{2}}$

Step 3.6.5.2

Factor $14$ out of $112$ .

$\frac{14 x^{2} + 14 \cdot 8}{{(x + 8)}^{2} {(x - 1)}^{2}}$

Step 3.6.5.3

Factor $14$ out of $14 x^{2} + 14 \cdot 8$ .

$\frac{14 (x^{2} + 8)}{{(x + 8)}^{2} {(x - 1)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = \frac{14 (x^{2} + 8)}{{(x + 8)}^{2} {(x - 1)}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{14 (x^{2} + 8)}{{(x + 8)}^{2} {(x - 1)}^{2}}$