Calculus Examples

$f = (10 x^{5} + 6 + 7 x^{- 5}) (- 8 x^{3} - 4)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (f) = \frac{d}{d x} ((10 x^{5} + 6 + 7 x^{- 5}) (- 8 x^{3} - 4))$

Step 2

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

$f'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = 10 x^{5} + 6 + 7 x^{- 5}$ and $g (x) = - 8 x^{3} - 4$ .

$(10 x^{5} + 6 + 7 x^{- 5}) \frac{d}{d x} [- 8 x^{3} - 4] + (- 8 x^{3} - 4) \frac{d}{d x} [10 x^{5} + 6 + 7 x^{- 5}]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $- 8 x^{3} - 4$ with respect to $x$ is $\frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 4]$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (\frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [- 4]) + (- 8 x^{3} - 4) \frac{d}{d x} [10 x^{5} + 6 + 7 x^{- 5}]$

Step 3.2.2

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4]) + (- 8 x^{3} - 4) \frac{d}{d x} [10 x^{5} + 6 + 7 x^{- 5}]$

Step 3.2.3

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

$(10 x^{5} + 6 + 7 x^{- 5}) (- 8 (3 x^{2}) + \frac{d}{d x} [- 4]) + (- 8 x^{3} - 4) \frac{d}{d x} [10 x^{5} + 6 + 7 x^{- 5}]$

Step 3.2.4

Multiply $3$ by $- 8$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2} + \frac{d}{d x} [- 4]) + (- 8 x^{3} - 4) \frac{d}{d x} [10 x^{5} + 6 + 7 x^{- 5}]$

Step 3.2.5

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

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2} + 0) + (- 8 x^{3} - 4) \frac{d}{d x} [10 x^{5} + 6 + 7 x^{- 5}]$

Step 3.2.6

Add $- 24 x^{2}$ and $0$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) \frac{d}{d x} [10 x^{5} + 6 + 7 x^{- 5}]$

Step 3.2.7

By the Sum Rule, the derivative of $10 x^{5} + 6 + 7 x^{- 5}$ with respect to $x$ is $\frac{d}{d x} [10 x^{5}] + \frac{d}{d x} [6] + \frac{d}{d x} [7 x^{- 5}]$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (\frac{d}{d x} [10 x^{5}] + \frac{d}{d x} [6] + \frac{d}{d x} [7 x^{- 5}])$

Step 3.2.8

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{5}$ with respect to $x$ is $10 \frac{d}{d x} [x^{5}]$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (10 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [6] + \frac{d}{d x} [7 x^{- 5}])$

Step 3.2.9

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

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (10 (5 x^{4}) + \frac{d}{d x} [6] + \frac{d}{d x} [7 x^{- 5}])$

Step 3.2.10

Multiply $5$ by $10$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} + \frac{d}{d x} [6] + \frac{d}{d x} [7 x^{- 5}])$

Step 3.2.11

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} + 0 + \frac{d}{d x} [7 x^{- 5}])$

Step 3.2.12

Add $50 x^{4}$ and $0$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} + \frac{d}{d x} [7 x^{- 5}])$

Step 3.2.13

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{- 5}$ with respect to $x$ is $7 \frac{d}{d x} [x^{- 5}]$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} + 7 \frac{d}{d x} [x^{- 5}])$

Step 3.2.14

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

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} + 7 (- 5 x^{- 6}))$

Step 3.2.15

Multiply $- 5$ by $7$ .

$(10 x^{5} + 6 + 7 x^{- 5}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 x^{- 6})$

Step 3.3

Simplify.

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Step 3.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(10 x^{5} + 6 + 7 \frac{1}{x^{5}}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 x^{- 6})$

Step 3.3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(10 x^{5} + 6 + 7 \frac{1}{x^{5}}) (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.3

Apply the distributive property.

$10 x^{5} (- 24 x^{2}) + 6 (- 24 x^{2}) + 7 \frac{1}{x^{5}} (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4

Combine terms.

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Step 3.3.4.1

Multiply $- 24$ by $10$ .

$- 240 x^{5} x^{2} + 6 (- 24 x^{2}) + 7 \frac{1}{x^{5}} (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.2

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

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Step 3.3.4.2.1

Move $x^{2}$ .

$- 240 (x^{2} x^{5}) + 6 (- 24 x^{2}) + 7 \frac{1}{x^{5}} (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.2.2

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

$- 240 x^{2 + 5} + 6 (- 24 x^{2}) + 7 \frac{1}{x^{5}} (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.2.3

Add $2$ and $5$ .

$- 240 x^{7} + 6 (- 24 x^{2}) + 7 \frac{1}{x^{5}} (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.3

Multiply $- 24$ by $6$ .

$- 240 x^{7} - 144 x^{2} + 7 \frac{1}{x^{5}} (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.4

Combine $7$ and $\frac{1}{x^{5}}$ .

$- 240 x^{7} - 144 x^{2} + \frac{7}{x^{5}} (- 24 x^{2}) + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.5

Combine $- 24$ and $\frac{7}{x^{5}}$ .

$- 240 x^{7} - 144 x^{2} + \frac{- 24 \cdot 7}{x^{5}} x^{2} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.6

Multiply $- 24$ by $7$ .

$- 240 x^{7} - 144 x^{2} + \frac{- 168}{x^{5}} x^{2} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.7

Combine $\frac{- 168}{x^{5}}$ and $x^{2}$ .

$- 240 x^{7} - 144 x^{2} + \frac{- 168 x^{2}}{x^{5}} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.8

Cancel the common factor of $x^{2}$ and $x^{5}$ .

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Step 3.3.4.8.1

Factor $x^{2}$ out of $- 168 x^{2}$ .

$- 240 x^{7} - 144 x^{2} + \frac{x^{2} \cdot - 168}{x^{5}} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.8.2

Cancel the common factors.

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Step 3.3.4.8.2.1

Factor $x^{2}$ out of $x^{5}$ .

$- 240 x^{7} - 144 x^{2} + \frac{x^{2} \cdot - 168}{x^{2} x^{3}} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.8.2.2

Cancel the common factor.

$- 240 x^{7} - 144 x^{2} + \frac{x^{2} \cdot - 168}{x^{2} x^{3}} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.8.2.3

Rewrite the expression.

$- 240 x^{7} - 144 x^{2} + \frac{- 168}{x^{3}} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.9

Move the negative in front of the fraction.

$- 240 x^{7} - 144 x^{2} - \frac{168}{x^{3}} + (- 8 x^{3} - 4) (50 x^{4} - 35 \frac{1}{x^{6}})$

Step 3.3.4.10

Combine $- 35$ and $\frac{1}{x^{6}}$ .

$- 240 x^{7} - 144 x^{2} - \frac{168}{x^{3}} + (- 8 x^{3} - 4) (50 x^{4} + \frac{- 35}{x^{6}})$

Step 3.3.4.11

Move the negative in front of the fraction.

$- 240 x^{7} - 144 x^{2} - \frac{168}{x^{3}} + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}})$

Step 3.3.4.12

To write $(- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}})$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$- 240 x^{7} - 144 x^{2} - \frac{168}{x^{3}} + \frac{(- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.13

Combine the numerators over the common denominator.

$- 240 x^{7} - 144 x^{2} + \frac{- 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.14

To write $- 240 x^{7}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$- 144 x^{2} - 240 x^{7} \cdot \frac{x^{3}}{x^{3}} + \frac{- 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.15

Combine $- 240 x^{7}$ and $\frac{x^{3}}{x^{3}}$ .

$- 144 x^{2} + \frac{- 240 x^{7} x^{3}}{x^{3}} + \frac{- 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.16

Combine the numerators over the common denominator.

$- 144 x^{2} + \frac{- 240 x^{7} x^{3} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.17

Multiply $x^{7}$ by $x^{3}$ by adding the exponents.

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Step 3.3.4.17.1

Move $x^{3}$ .

$- 144 x^{2} + \frac{- 240 (x^{3} x^{7}) - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.17.2

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

$- 144 x^{2} + \frac{- 240 x^{3 + 7} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.17.3

Add $3$ and $7$ .

$- 144 x^{2} + \frac{- 240 x^{10} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.18

To write $- 144 x^{2}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$- 144 x^{2} \cdot \frac{x^{3}}{x^{3}} + \frac{- 240 x^{10} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.19

Combine $- 144 x^{2}$ and $\frac{x^{3}}{x^{3}}$ .

$\frac{- 144 x^{2} x^{3}}{x^{3}} + \frac{- 240 x^{10} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.20

Combine the numerators over the common denominator.

$\frac{- 144 x^{2} x^{3} - 240 x^{10} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.21

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 3.3.4.21.1

Move $x^{3}$ .

$\frac{- 144 (x^{3} x^{2}) - 240 x^{10} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.21.2

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

$\frac{- 144 x^{3 + 2} - 240 x^{10} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.4.21.3

Add $3$ and $2$ .

$\frac{- 144 x^{5} - 240 x^{10} - 168 + (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) x^{3}}{x^{3}}$

Step 3.3.5

Reorder terms.

$\frac{- 240 x^{10} - 144 x^{5} + x^{3} (- 8 x^{3} - 4) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6

Simplify the numerator.

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Step 3.3.6.1

Apply the distributive property.

$\frac{- 240 x^{10} - 144 x^{5} + (x^{3} (- 8 x^{3}) + x^{3} \cdot - 4) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.2

Rewrite using the commutative property of multiplication.

$\frac{- 240 x^{10} - 144 x^{5} + (- 8 x^{3} x^{3} + x^{3} \cdot - 4) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.3

Move $- 4$ to the left of $x^{3}$ .

$\frac{- 240 x^{10} - 144 x^{5} + (- 8 x^{3} x^{3} - 4 \cdot x^{3}) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.4

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

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Step 3.3.6.4.1

Move $x^{3}$ .

$\frac{- 240 x^{10} - 144 x^{5} + (- 8 (x^{3} x^{3}) - 4 \cdot x^{3}) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.4.2

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

$\frac{- 240 x^{10} - 144 x^{5} + (- 8 x^{3 + 3} - 4 \cdot x^{3}) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.4.3

Add $3$ and $3$ .

$\frac{- 240 x^{10} - 144 x^{5} + (- 8 x^{6} - 4 \cdot x^{3}) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

$\frac{- 240 x^{10} - 144 x^{5} + (- 8 x^{6} - 4 x^{3}) (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.5

Expand $(- 8 x^{6} - 4 x^{3}) (50 x^{4} - \frac{35}{x^{6}})$ using the FOIL Method.

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Step 3.3.6.5.1

Apply the distributive property.

$\frac{- 240 x^{10} - 144 x^{5} - 8 x^{6} (50 x^{4} - \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.5.2

Apply the distributive property.

$\frac{- 240 x^{10} - 144 x^{5} - 8 x^{6} (50 x^{4}) - 8 x^{6} (- \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4} - \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.5.3

Apply the distributive property.

$\frac{- 240 x^{10} - 144 x^{5} - 8 x^{6} (50 x^{4}) - 8 x^{6} (- \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6

Simplify each term.

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Step 3.3.6.6.1

Rewrite using the commutative property of multiplication.

$\frac{- 240 x^{10} - 144 x^{5} - 8 \cdot 50 x^{6} x^{4} - 8 x^{6} (- \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.2

Multiply $x^{6}$ by $x^{4}$ by adding the exponents.

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Step 3.3.6.6.2.1

Move $x^{4}$ .

$\frac{- 240 x^{10} - 144 x^{5} - 8 \cdot 50 (x^{4} x^{6}) - 8 x^{6} (- \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.2.2

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

$\frac{- 240 x^{10} - 144 x^{5} - 8 \cdot 50 x^{4 + 6} - 8 x^{6} (- \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.2.3

Add $4$ and $6$ .

$\frac{- 240 x^{10} - 144 x^{5} - 8 \cdot 50 x^{10} - 8 x^{6} (- \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.3

Multiply $- 8$ by $50$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} - 8 x^{6} (- \frac{35}{x^{6}}) - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.4

Cancel the common factor of $x^{6}$ .

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Step 3.3.6.6.4.1

Move the leading negative in $- \frac{35}{x^{6}}$ into the numerator.

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} - 8 x^{6} \frac{- 35}{x^{6}} - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.4.2

Factor $x^{6}$ out of $- 8 x^{6}$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + x^{6} \cdot - 8 \frac{- 35}{x^{6}} - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.4.3

Cancel the common factor.

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + x^{6} \cdot - 8 \frac{- 35}{x^{6}} - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.4.4

Rewrite the expression.

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} - 8 \cdot - 35 - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.5

Multiply $- 8$ by $- 35$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 4 x^{3} (50 x^{4}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.6

Rewrite using the commutative property of multiplication.

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 4 \cdot 50 x^{3} x^{4} - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.7

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 3.3.6.6.7.1

Move $x^{4}$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 4 \cdot 50 (x^{4} x^{3}) - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.7.2

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

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 4 \cdot 50 x^{4 + 3} - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.7.3

Add $4$ and $3$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 4 \cdot 50 x^{7} - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.8

Multiply $- 4$ by $50$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} - 4 x^{3} (- \frac{35}{x^{6}}) - 168}{x^{3}}$

Step 3.3.6.6.9

Cancel the common factor of $x^{3}$ .

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Step 3.3.6.6.9.1

Move the leading negative in $- \frac{35}{x^{6}}$ into the numerator.

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} - 4 x^{3} \frac{- 35}{x^{6}} - 168}{x^{3}}$

Step 3.3.6.6.9.2

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

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} + x^{3} \cdot - 4 \frac{- 35}{x^{6}} - 168}{x^{3}}$

Step 3.3.6.6.9.3

Factor $x^{3}$ out of $x^{6}$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} + x^{3} \cdot - 4 \frac{- 35}{x^{3} x^{3}} - 168}{x^{3}}$

Step 3.3.6.6.9.4

Cancel the common factor.

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} + x^{3} \cdot - 4 \frac{- 35}{x^{3} x^{3}} - 168}{x^{3}}$

Step 3.3.6.6.9.5

Rewrite the expression.

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} - 4 \frac{- 35}{x^{3}} - 168}{x^{3}}$

Step 3.3.6.6.10

Combine $- 4$ and $\frac{- 35}{x^{3}}$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} + \frac{- 4 \cdot - 35}{x^{3}} - 168}{x^{3}}$

Step 3.3.6.6.11

Multiply $- 4$ by $- 35$ .

$\frac{- 240 x^{10} - 144 x^{5} - 400 x^{10} + 280 - 200 x^{7} + \frac{140}{x^{3}} - 168}{x^{3}}$

Step 3.3.6.7

Subtract $400 x^{10}$ from $- 240 x^{10}$ .

$\frac{- 640 x^{10} - 144 x^{5} + 280 - 200 x^{7} + \frac{140}{x^{3}} - 168}{x^{3}}$

Step 3.3.6.8

Subtract $168$ from $280$ .

$\frac{- 640 x^{10} - 144 x^{5} - 200 x^{7} + \frac{140}{x^{3}} + 112}{x^{3}}$

Step 3.3.6.9

Reorder terms.

$\frac{- 640 x^{10} - 200 x^{7} - 144 x^{5} + \frac{140}{x^{3}} + 112}{x^{3}}$

Step 3.3.6.10

Factor $4$ out of $- 640 x^{10} - 200 x^{7} - 144 x^{5} + \frac{140}{x^{3}} + 112$ .

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Step 3.3.6.10.1

Factor $4$ out of $- 640 x^{10}$ .

$\frac{4 (- 160 x^{10}) - 200 x^{7} - 144 x^{5} + \frac{140}{x^{3}} + 112}{x^{3}}$

Step 3.3.6.10.2

Factor $4$ out of $- 200 x^{7}$ .

$\frac{4 (- 160 x^{10}) + 4 (- 50 x^{7}) - 144 x^{5} + \frac{140}{x^{3}} + 112}{x^{3}}$

Step 3.3.6.10.3

Factor $4$ out of $- 144 x^{5}$ .

$\frac{4 (- 160 x^{10}) + 4 (- 50 x^{7}) + 4 (- 36 x^{5}) + \frac{140}{x^{3}} + 112}{x^{3}}$

Step 3.3.6.10.4

Factor $4$ out of $\frac{140}{x^{3}}$ .

$\frac{4 (- 160 x^{10}) + 4 (- 50 x^{7}) + 4 (- 36 x^{5}) + 4 \frac{35}{x^{3}} + 112}{x^{3}}$

Step 3.3.6.10.5

Factor $4$ out of $112$ .

$\frac{4 (- 160 x^{10}) + 4 (- 50 x^{7}) + 4 (- 36 x^{5}) + 4 \frac{35}{x^{3}} + 4 \cdot 28}{x^{3}}$

Step 3.3.6.10.6

Factor $4$ out of $4 (- 160 x^{10}) + 4 (- 50 x^{7})$ .

$\frac{4 (- 160 x^{10} - 50 x^{7}) + 4 (- 36 x^{5}) + 4 \frac{35}{x^{3}} + 4 \cdot 28}{x^{3}}$

Step 3.3.6.10.7

Factor $4$ out of $4 (- 160 x^{10} - 50 x^{7}) + 4 (- 36 x^{5})$ .

$\frac{4 (- 160 x^{10} - 50 x^{7} - 36 x^{5}) + 4 \frac{35}{x^{3}} + 4 \cdot 28}{x^{3}}$

Step 3.3.6.10.8

Factor $4$ out of $4 (- 160 x^{10} - 50 x^{7} - 36 x^{5}) + 4 \frac{35}{x^{3}}$ .

$\frac{4 (- 160 x^{10} - 50 x^{7} - 36 x^{5} + \frac{35}{x^{3}}) + 4 \cdot 28}{x^{3}}$

Step 3.3.6.10.9

Factor $4$ out of $4 (- 160 x^{10} - 50 x^{7} - 36 x^{5} + \frac{35}{x^{3}}) + 4 \cdot 28$ .

$\frac{4 (- 160 x^{10} - 50 x^{7} - 36 x^{5} + \frac{35}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.11

To write $- 160 x^{10}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{4 (- 50 x^{7} - 36 x^{5} - 160 x^{10} \cdot \frac{x^{3}}{x^{3}} + \frac{35}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.12

Combine $- 160 x^{10}$ and $\frac{x^{3}}{x^{3}}$ .

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{- 160 x^{10} x^{3}}{x^{3}} + \frac{35}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.13

Combine the numerators over the common denominator.

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{- 160 x^{10} x^{3} + 35}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.14

Simplify the numerator.

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Step 3.3.6.14.1

Factor $5$ out of $- 160 x^{10} x^{3} + 35$ .

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Step 3.3.6.14.1.1

Factor $5$ out of $- 160 x^{10} x^{3}$ .

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{5 (- 32 x^{10} x^{3}) + 35}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.14.1.2

Factor $5$ out of $35$ .

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{5 (- 32 x^{10} x^{3}) + 5 (7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.14.1.3

Factor $5$ out of $5 (- 32 x^{10} x^{3}) + 5 (7)$ .

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{5 (- 32 x^{10} x^{3} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.14.2

Multiply $x^{10}$ by $x^{3}$ by adding the exponents.

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Step 3.3.6.14.2.1

Move $x^{3}$ .

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{5 (- 32 (x^{3} x^{10}) + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.14.2.2

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

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{5 (- 32 x^{3 + 10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.14.2.3

Add $3$ and $10$ .

$\frac{4 (- 50 x^{7} - 36 x^{5} + \frac{5 (- 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.15

To write $- 50 x^{7}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{4 (- 36 x^{5} - 50 x^{7} \cdot \frac{x^{3}}{x^{3}} + \frac{5 (- 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.16

Combine $- 50 x^{7}$ and $\frac{x^{3}}{x^{3}}$ .

$\frac{4 (- 36 x^{5} + \frac{- 50 x^{7} x^{3}}{x^{3}} + \frac{5 (- 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.17

Combine the numerators over the common denominator.

$\frac{4 (- 36 x^{5} + \frac{- 50 x^{7} x^{3} + 5 (- 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.18

Simplify the numerator.

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Step 3.3.6.18.1

Factor $5$ out of $- 50 x^{7} x^{3} + 5 (- 32 x^{13} + 7)$ .

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Step 3.3.6.18.1.1

Factor $5$ out of $- 50 x^{7} x^{3}$ .

$\frac{4 (- 36 x^{5} + \frac{5 (- 10 x^{7} x^{3}) + 5 (- 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.18.1.2

Factor $5$ out of $5 (- 10 x^{7} x^{3}) + 5 (- 32 x^{13} + 7)$ .

$\frac{4 (- 36 x^{5} + \frac{5 (- 10 x^{7} x^{3} - 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.18.2

Multiply $x^{7}$ by $x^{3}$ by adding the exponents.

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Step 3.3.6.18.2.1

Move $x^{3}$ .

$\frac{4 (- 36 x^{5} + \frac{5 (- 10 (x^{3} x^{7}) - 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.18.2.2

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

$\frac{4 (- 36 x^{5} + \frac{5 (- 10 x^{3 + 7} - 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.18.2.3

Add $3$ and $7$ .

$\frac{4 (- 36 x^{5} + \frac{5 (- 10 x^{10} - 32 x^{13} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.18.3

Reorder terms.

$\frac{4 (- 36 x^{5} + \frac{5 (- 32 x^{13} - 10 x^{10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.19

To write $- 36 x^{5}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{4 (- 36 x^{5} \cdot \frac{x^{3}}{x^{3}} + \frac{5 (- 32 x^{13} - 10 x^{10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.20

Combine $- 36 x^{5}$ and $\frac{x^{3}}{x^{3}}$ .

$\frac{4 (\frac{- 36 x^{5} x^{3}}{x^{3}} + \frac{5 (- 32 x^{13} - 10 x^{10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.21

Combine the numerators over the common denominator.

$\frac{4 (\frac{- 36 x^{5} x^{3} + 5 (- 32 x^{13} - 10 x^{10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22

Simplify the numerator.

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Step 3.3.6.22.1

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

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Step 3.3.6.22.1.1

Move $x^{3}$ .

$\frac{4 (\frac{- 36 (x^{3} x^{5}) + 5 (- 32 x^{13} - 10 x^{10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22.1.2

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

$\frac{4 (\frac{- 36 x^{3 + 5} + 5 (- 32 x^{13} - 10 x^{10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22.1.3

Add $3$ and $5$ .

$\frac{4 (\frac{- 36 x^{8} + 5 (- 32 x^{13} - 10 x^{10} + 7)}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22.2

Apply the distributive property.

$\frac{4 (\frac{- 36 x^{8} + 5 (- 32 x^{13}) + 5 (- 10 x^{10}) + 5 \cdot 7}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22.3

Simplify.

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Step 3.3.6.22.3.1

Multiply $- 32$ by $5$ .

$\frac{4 (\frac{- 36 x^{8} - 160 x^{13} + 5 (- 10 x^{10}) + 5 \cdot 7}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22.3.2

Multiply $- 10$ by $5$ .

$\frac{4 (\frac{- 36 x^{8} - 160 x^{13} - 50 x^{10} + 5 \cdot 7}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22.3.3

Multiply $5$ by $7$ .

$\frac{4 (\frac{- 36 x^{8} - 160 x^{13} - 50 x^{10} + 35}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.22.4

Reorder terms.

$\frac{4 (\frac{- 160 x^{13} - 50 x^{10} - 36 x^{8} + 35}{x^{3}} + 28)}{x^{3}}$

Step 3.3.6.23

To write $28$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{4 (\frac{- 160 x^{13} - 50 x^{10} - 36 x^{8} + 35}{x^{3}} + \frac{28 x^{3}}{x^{3}})}{x^{3}}$

Step 3.3.6.24

Combine the numerators over the common denominator.

$\frac{4 \frac{- 160 x^{13} - 50 x^{10} - 36 x^{8} + 35 + 28 x^{3}}{x^{3}}}{x^{3}}$

Step 3.3.6.25

Reorder terms.

$\frac{4 \frac{- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35}{x^{3}}}{x^{3}}$

Step 3.3.7

Combine $4$ and $\frac{- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35}{x^{3}}$ .

$\frac{\frac{4 (- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35)}{x^{3}}}{x^{3}}$

Step 3.3.8

Multiply the numerator by the reciprocal of the denominator.

$\frac{4 (- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35)}{x^{3}} \cdot \frac{1}{x^{3}}$

Step 3.3.9

Combine.

$\frac{4 (- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35) \cdot 1}{x^{3} x^{3}}$

Step 3.3.10

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

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Step 3.3.10.1

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

$\frac{4 (- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35) \cdot 1}{x^{3 + 3}}$

Step 3.3.10.2

Add $3$ and $3$ .

$\frac{4 (- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35) \cdot 1}{x^{6}}$

Step 3.3.11

Multiply $4 (- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35)$ by $1$ .

$\frac{4 (- 160 x^{13} - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35)}{x^{6}}$

Step 3.3.12

Factor $- 1$ out of $- 160 x^{13}$ .

$\frac{4 (- (160 x^{13}) - 50 x^{10} - 36 x^{8} + 28 x^{3} + 35)}{x^{6}}$

Step 3.3.13

Factor $- 1$ out of $- 50 x^{10}$ .

$\frac{4 (- (160 x^{13}) - (50 x^{10}) - 36 x^{8} + 28 x^{3} + 35)}{x^{6}}$

Step 3.3.14

Factor $- 1$ out of $- (160 x^{13}) - (50 x^{10})$ .

$\frac{4 (- (160 x^{13} + 50 x^{10}) - 36 x^{8} + 28 x^{3} + 35)}{x^{6}}$

Step 3.3.15

Factor $- 1$ out of $- 36 x^{8}$ .

$\frac{4 (- (160 x^{13} + 50 x^{10}) - (36 x^{8}) + 28 x^{3} + 35)}{x^{6}}$

Step 3.3.16

Factor $- 1$ out of $- (160 x^{13} + 50 x^{10}) - (36 x^{8})$ .

$\frac{4 (- (160 x^{13} + 50 x^{10} + 36 x^{8}) + 28 x^{3} + 35)}{x^{6}}$

Step 3.3.17

Factor $- 1$ out of $28 x^{3}$ .

$\frac{4 (- (160 x^{13} + 50 x^{10} + 36 x^{8}) - (- 28 x^{3}) + 35)}{x^{6}}$

Step 3.3.18

Factor $- 1$ out of $- (160 x^{13} + 50 x^{10} + 36 x^{8}) - (- 28 x^{3})$ .

$\frac{4 (- (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3}) + 35)}{x^{6}}$

Step 3.3.19

Rewrite $35$ as $- 1 (- 35)$ .

$\frac{4 (- (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3}) - 1 (- 35))}{x^{6}}$

Step 3.3.20

Factor $- 1$ out of $- (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3}) - 1 (- 35)$ .

$\frac{4 (- (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3} - 35))}{x^{6}}$

Step 3.3.21

Rewrite $- (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3} - 35)$ as $- 1 (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3} - 35)$ .

$\frac{4 (- 1 (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3} - 35))}{x^{6}}$

Step 3.3.22

Move the negative in front of the fraction.

$- \frac{4 (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3} - 35)}{x^{6}}$

Step 4

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

$f' = - \frac{4 (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3} - 35)}{x^{6}}$

Step 5

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

$\frac{d f}{d x} = - \frac{4 (160 x^{13} + 50 x^{10} + 36 x^{8} - 28 x^{3} - 35)}{x^{6}}$