Calculus Examples

$y = \frac{(x^{6} + 4) (b x - 4)}{x^{5}}$

Step 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 $f (x) = (x^{6} + 4) (b x - 4)$ and $g (x) = x^{5}$ .

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

Step 2

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 2.1

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

$\frac{x^{5} \frac{d}{d x} [(x^{6} + 4) (b x - 4)] - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{5 \cdot 2}}$

Step 2.2

Multiply $5$ by $2$ .

$\frac{x^{5} \frac{d}{d x} [(x^{6} + 4) (b x - 4)] - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{6} + 4$ and $g (x) = b x - 4$ .

$\frac{x^{5} ((x^{6} + 4) \frac{d}{d x} [b x - 4] + (b x - 4) \frac{d}{d x} [x^{6} + 4]) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4

Differentiate.

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Step 4.1

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

$\frac{x^{5} ((x^{6} + 4) (\frac{d}{d x} [b x] + \frac{d}{d x} [- 4]) + (b x - 4) \frac{d}{d x} [x^{6} + 4]) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.2

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

$\frac{x^{5} ((x^{6} + 4) (b \frac{d}{d x} [x] + \frac{d}{d x} [- 4]) + (b x - 4) \frac{d}{d x} [x^{6} + 4]) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.3

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

$\frac{x^{5} ((x^{6} + 4) (b \cdot 1 + \frac{d}{d x} [- 4]) + (b x - 4) \frac{d}{d x} [x^{6} + 4]) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.4

Multiply $b$ by $1$ .

$\frac{x^{5} ((x^{6} + 4) (b + \frac{d}{d x} [- 4]) + (b x - 4) \frac{d}{d x} [x^{6} + 4]) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.5

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{x^{5} ((x^{6} + 4) (b + 0) + (b x - 4) \frac{d}{d x} [x^{6} + 4]) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.6

Add $b$ and $0$ .

$\frac{x^{5} ((x^{6} + 4) b + (b x - 4) \frac{d}{d x} [x^{6} + 4]) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.7

By the Sum Rule, the derivative of $x^{6} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{6}] + \frac{d}{d x} [4]$ .

$\frac{x^{5} ((x^{6} + 4) b + (b x - 4) (\frac{d}{d x} [x^{6}] + \frac{d}{d x} [4])) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{x^{5} ((x^{6} + 4) b + (b x - 4) (6 x^{5} + \frac{d}{d x} [4])) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.9

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{x^{5} ((x^{6} + 4) b + (b x - 4) (6 x^{5} + 0)) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.10

Simplify the expression.

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Step 4.10.1

Add $6 x^{5}$ and $0$ .

$\frac{x^{5} ((x^{6} + 4) b + (b x - 4) (6 x^{5})) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.10.2

Move $6$ to the left of $b x - 4$ .

$\frac{x^{5} ((x^{6} + 4) b + 6 \cdot (b x - 4) x^{5}) - (x^{6} + 4) (b x - 4) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 4.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{x^{5} ((x^{6} + 4) b + 6 (b x - 4) x^{5}) - (x^{6} + 4) (b x - 4) (5 x^{4})}{x^{10}}$

Step 4.12

Simplify with factoring out.

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Step 4.12.1

Multiply $5$ by $- 1$ .

$\frac{x^{5} ((x^{6} + 4) b + 6 (b x - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4) x^{4}}{x^{10}}$

Step 4.12.2

Factor $x^{4}$ out of $x^{5} ((x^{6} + 4) b + 6 (b x - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4) x^{4}$ .

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Step 4.12.2.1

Factor $x^{4}$ out of $x^{5} ((x^{6} + 4) b + 6 (b x - 4) x^{5})$ .

$\frac{x^{4} (x ((x^{6} + 4) b + 6 (b x - 4) x^{5})) - 5 (x^{6} + 4) (b x - 4) x^{4}}{x^{10}}$

Step 4.12.2.2

Factor $x^{4}$ out of $- 5 (x^{6} + 4) (b x - 4) x^{4}$ .

$\frac{x^{4} (x ((x^{6} + 4) b + 6 (b x - 4) x^{5})) + x^{4} (- 5 (x^{6} + 4) (b x - 4))}{x^{10}}$

Step 4.12.2.3

Factor $x^{4}$ out of $x^{4} (x ((x^{6} + 4) b + 6 (b x - 4) x^{5})) + x^{4} (- 5 (x^{6} + 4) (b x - 4))$ .

$\frac{x^{4} (x ((x^{6} + 4) b + 6 (b x - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4))}{x^{10}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x^{4}$ out of $x^{10}$ .

$\frac{x^{4} (x ((x^{6} + 4) b + 6 (b x - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4))}{x^{4} x^{6}}$

Step 5.2

Cancel the common factor.

$\frac{x^{4} (x ((x^{6} + 4) b + 6 (b x - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4))}{x^{4} x^{6}}$

Step 5.3

Rewrite the expression.

$\frac{x ((x^{6} + 4) b + 6 (b x - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4)}{x^{6}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{x (x^{6} b + 4 b + 6 (b x - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4)}{x^{6}}$

Step 6.2

Apply the distributive property.

$\frac{x (x^{6} b + 4 b + (6 (b x) + 6 \cdot - 4) x^{5}) - 5 (x^{6} + 4) (b x - 4)}{x^{6}}$

Step 6.3

Apply the distributive property.

$\frac{x (x^{6} b + 4 b + 6 (b x) x^{5} + 6 \cdot - 4 x^{5}) - 5 (x^{6} + 4) (b x - 4)}{x^{6}}$

Step 6.4

Apply the distributive property.

$\frac{x (x^{6} b) + x (4 b) + x (6 (b x) x^{5}) + x (6 \cdot - 4 x^{5}) - 5 (x^{6} + 4) (b x - 4)}{x^{6}}$

Step 6.5

Apply the distributive property.

$\frac{x (x^{6} b) + x (4 b) + x (6 (b x) x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 6.6.1.1.1

Move $x^{6}$ .

$\frac{x^{6} x b + x (4 b) + x (6 (b x) x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.1.2

Multiply $x^{6}$ by $x$ .

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Step 6.6.1.1.2.1

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

$\frac{x^{6} x^{1} b + x (4 b) + x (6 (b x) x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.1.2.2

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

$\frac{x^{6 + 1} b + x (4 b) + x (6 (b x) x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.1.3

Add $6$ and $1$ .

$\frac{x^{7} b + x (4 b) + x (6 (b x) x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.2

Rewrite using the commutative property of multiplication.

$\frac{x^{7} b + 4 x b + x (6 (b x) x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.3

Rewrite using the commutative property of multiplication.

$\frac{x^{7} b + 4 x b + 6 x (b x \cdot x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.4

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

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Step 6.6.1.4.1

Move $x$ .

$\frac{x^{7} b + 4 x b + 6 (x \cdot x) (b x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.4.2

Multiply $x$ by $x$ .

$\frac{x^{7} b + 4 x b + 6 x^{2} (b x^{5}) + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.5

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

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Step 6.6.1.5.1

Move $x^{5}$ .

$\frac{x^{7} b + 4 x b + 6 (x^{5} x^{2}) b + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.5.2

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

$\frac{x^{7} b + 4 x b + 6 x^{5 + 2} b + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.5.3

Add $5$ and $2$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b + x (6 \cdot - 4 x^{5}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.6

Rewrite using the commutative property of multiplication.

$\frac{x^{7} b + 4 x b + 6 x^{7} b + 6 \cdot - 4 x \cdot x^{5} + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.7

Multiply $x$ by $x^{5}$ by adding the exponents.

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Step 6.6.1.7.1

Move $x^{5}$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b + 6 \cdot - 4 (x^{5} x) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.7.2

Multiply $x^{5}$ by $x$ .

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Step 6.6.1.7.2.1

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

$\frac{x^{7} b + 4 x b + 6 x^{7} b + 6 \cdot - 4 (x^{5} x^{1}) + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.7.2.2

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

$\frac{x^{7} b + 4 x b + 6 x^{7} b + 6 \cdot - 4 x^{5 + 1} + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.7.3

Add $5$ and $1$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b + 6 \cdot - 4 x^{6} + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.8

Multiply $6$ by $- 4$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} + (- 5 x^{6} - 5 \cdot 4) (b x - 4)}{x^{6}}$

Step 6.6.1.9

Multiply $- 5$ by $4$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} + (- 5 x^{6} - 20) (b x - 4)}{x^{6}}$

Step 6.6.1.10

Expand $(- 5 x^{6} - 20) (b x - 4)$ using the FOIL Method.

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Step 6.6.1.10.1

Apply the distributive property.

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 x^{6} (b x - 4) - 20 (b x - 4)}{x^{6}}$

Step 6.6.1.10.2

Apply the distributive property.

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 x^{6} (b x) - 5 x^{6} \cdot - 4 - 20 (b x - 4)}{x^{6}}$

Step 6.6.1.10.3

Apply the distributive property.

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 x^{6} (b x) - 5 x^{6} \cdot - 4 - 20 (b x) - 20 \cdot - 4}{x^{6}}$

Step 6.6.1.11

Simplify each term.

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Step 6.6.1.11.1

Multiply $x^{6}$ by $x$ by adding the exponents.

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Step 6.6.1.11.1.1

Move $x$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 (x \cdot x^{6}) b - 5 x^{6} \cdot - 4 - 20 (b x) - 20 \cdot - 4}{x^{6}}$

Step 6.6.1.11.1.2

Multiply $x$ by $x^{6}$ .

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Step 6.6.1.11.1.2.1

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

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 (x^{1} x^{6}) b - 5 x^{6} \cdot - 4 - 20 (b x) - 20 \cdot - 4}{x^{6}}$

Step 6.6.1.11.1.2.2

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

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 x^{1 + 6} b - 5 x^{6} \cdot - 4 - 20 (b x) - 20 \cdot - 4}{x^{6}}$

Step 6.6.1.11.1.3

Add $1$ and $6$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 x^{7} b - 5 x^{6} \cdot - 4 - 20 (b x) - 20 \cdot - 4}{x^{6}}$

Step 6.6.1.11.2

Multiply $- 4$ by $- 5$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 x^{7} b + 20 x^{6} - 20 (b x) - 20 \cdot - 4}{x^{6}}$

Step 6.6.1.11.3

Multiply $- 20$ by $- 4$ .

$\frac{x^{7} b + 4 x b + 6 x^{7} b - 24 x^{6} - 5 x^{7} b + 20 x^{6} - 20 b x + 80}{x^{6}}$

Step 6.6.2

Add $x^{7} b$ and $6 x^{7} b$ .

$\frac{7 x^{7} b + 4 x b - 24 x^{6} - 5 x^{7} b + 20 x^{6} - 20 b x + 80}{x^{6}}$

Step 6.6.3

Subtract $5 x^{7} b$ from $7 x^{7} b$ .

$\frac{4 x b - 24 x^{6} + 2 x^{7} b + 20 x^{6} - 20 b x + 80}{x^{6}}$

Step 6.6.4

Subtract $20 b x$ from $4 x b$ .

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Step 6.6.4.1

Move $x$ .

$\frac{- 24 x^{6} + 2 x^{7} b + 20 x^{6} + 4 b x - 20 b x + 80}{x^{6}}$

Step 6.6.4.2

Subtract $20 b x$ from $4 b x$ .

$\frac{- 24 x^{6} + 2 x^{7} b + 20 x^{6} - 16 b x + 80}{x^{6}}$

Step 6.6.5

Add $- 24 x^{6}$ and $20 x^{6}$ .

$\frac{- 4 x^{6} + 2 x^{7} b - 16 b x + 80}{x^{6}}$

Step 6.7

Factor $2$ out of $- 4 x^{6} + 2 x^{7} b - 16 b x + 80$ .

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Step 6.7.1

Factor $2$ out of $- 4 x^{6}$ .

$\frac{2 (- 2 x^{6}) + 2 x^{7} b - 16 b x + 80}{x^{6}}$

Step 6.7.2

Factor $2$ out of $2 x^{7} b$ .

$\frac{2 (- 2 x^{6}) + 2 (x^{7} b) - 16 b x + 80}{x^{6}}$

Step 6.7.3

Factor $2$ out of $- 16 b x$ .

$\frac{2 (- 2 x^{6}) + 2 (x^{7} b) + 2 (- 8 b x) + 80}{x^{6}}$

Step 6.7.4

Factor $2$ out of $80$ .

$\frac{2 (- 2 x^{6}) + 2 (x^{7} b) + 2 (- 8 b x) + 2 \cdot 40}{x^{6}}$

Step 6.7.5

Factor $2$ out of $2 (- 2 x^{6}) + 2 (x^{7} b)$ .

$\frac{2 (- 2 x^{6} + x^{7} b) + 2 (- 8 b x) + 2 \cdot 40}{x^{6}}$

Step 6.7.6

Factor $2$ out of $2 (- 2 x^{6} + x^{7} b) + 2 (- 8 b x)$ .

$\frac{2 (- 2 x^{6} + x^{7} b - 8 b x) + 2 \cdot 40}{x^{6}}$

Step 6.7.7

Factor $2$ out of $2 (- 2 x^{6} + x^{7} b - 8 b x) + 2 \cdot 40$ .

$\frac{2 (- 2 x^{6} + x^{7} b - 8 b x + 40)}{x^{6}}$

Step 6.8

Factor $- 1$ out of $- 2 x^{6}$ .

$\frac{2 (- (2 x^{6}) + x^{7} b - 8 b x + 40)}{x^{6}}$

Step 6.9

Factor $- 1$ out of $x^{7} b$ .

$\frac{2 (- (2 x^{6}) - (- x^{7} b) - 8 b x + 40)}{x^{6}}$

Step 6.10

Factor $- 1$ out of $- (2 x^{6}) - (- x^{7} b)$ .

$\frac{2 (- (2 x^{6} - x^{7} b) - 8 b x + 40)}{x^{6}}$

Step 6.11

Factor $- 1$ out of $- 8 b x$ .

$\frac{2 (- (2 x^{6} - x^{7} b) - (8 b x) + 40)}{x^{6}}$

Step 6.12

Factor $- 1$ out of $- (2 x^{6} - x^{7} b) - (8 b x)$ .

$\frac{2 (- (2 x^{6} - x^{7} b + 8 b x) + 40)}{x^{6}}$

Step 6.13

Rewrite $40$ as $- 1 (- 40)$ .

$\frac{2 (- (2 x^{6} - x^{7} b + 8 b x) - 1 (- 40))}{x^{6}}$

Step 6.14

Factor $- 1$ out of $- (2 x^{6} - x^{7} b + 8 b x) - 1 (- 40)$ .

$\frac{2 (- (2 x^{6} - x^{7} b + 8 b x - 40))}{x^{6}}$

Step 6.15

Rewrite $- (2 x^{6} - x^{7} b + 8 b x - 40)$ as $- 1 (2 x^{6} - x^{7} b + 8 b x - 40)$ .

$\frac{2 (- 1 (2 x^{6} - x^{7} b + 8 b x - 40))}{x^{6}}$

Step 6.16

Move the negative in front of the fraction.

$- \frac{2 (2 x^{6} - x^{7} b + 8 b x - 40)}{x^{6}}$