Calculus Examples

$\frac{d}{d x} (\frac{- 2 x^{4} - x^{- 2} - 4}{- 4 x^{7}})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{- 2 x^{4} - x^{- 2} - 4}{4 x^{7}}]$

Step 1.2

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{- 2 x^{4} - x^{- 2} - 4}{4 x^{7}}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [\frac{- 2 x^{4} - x^{- 2} - 4}{x^{7}}]$ .

$- \frac{1}{4} \frac{d}{d x} [\frac{- 2 x^{4} - x^{- 2} - 4}{x^{7}}]$

Step 2

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 $f (x) = - 2 x^{4} - x^{- 2} - 4$ and $g (x) = x^{7}$ .

$- \frac{1}{4} \cdot \frac{x^{7} \frac{d}{d x} [- 2 x^{4} - x^{- 2} - 4] - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{{(x^{7})}^{2}}$

Step 3

Differentiate.

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Step 3.1

Multiply the exponents in ${(x^{7})}^{2}$ .

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Step 3.1.1

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

$- \frac{1}{4} \cdot \frac{x^{7} \frac{d}{d x} [- 2 x^{4} - x^{- 2} - 4] - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{7 \cdot 2}}$

Step 3.1.2

Multiply $7$ by $2$ .

$- \frac{1}{4} \cdot \frac{x^{7} \frac{d}{d x} [- 2 x^{4} - x^{- 2} - 4] - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.2

By the Sum Rule, the derivative of $- 2 x^{4} - x^{- 2} - 4$ with respect to $x$ is $\frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- x^{- 2}] + \frac{d}{d x} [- 4]$ .

$- \frac{1}{4} \cdot \frac{x^{7} (\frac{d}{d x} [- 2 x^{4}] + \frac{d}{d x} [- x^{- 2}] + \frac{d}{d x} [- 4]) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{4}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{4}]$ .

$- \frac{1}{4} \cdot \frac{x^{7} (- 2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- x^{- 2}] + \frac{d}{d x} [- 4]) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$- \frac{1}{4} \cdot \frac{x^{7} (- 2 (4 x^{3}) + \frac{d}{d x} [- x^{- 2}] + \frac{d}{d x} [- 4]) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.5

Multiply $4$ by $- 2$ .

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} + \frac{d}{d x} [- x^{- 2}] + \frac{d}{d x} [- 4]) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{- 2}$ with respect to $x$ is $- \frac{d}{d x} [x^{- 2}]$ .

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} - \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [- 4]) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.7

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

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} - (- 2 x^{- 3}) + \frac{d}{d x} [- 4]) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.8

Multiply $- 2$ by $- 1$ .

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} + 2 x^{- 3} + \frac{d}{d x} [- 4]) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.9

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

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} + 2 x^{- 3} + 0) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.10

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

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} + 2 x^{- 3}) - (- 2 x^{4} - x^{- 2} - 4) \frac{d}{d x} [x^{7}]}{x^{14}}$

Step 3.11

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} + 2 x^{- 3}) - (- 2 x^{4} - x^{- 2} - 4) (7 x^{6})}{x^{14}}$

Step 3.12

Simplify with factoring out.

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Step 3.12.1

Multiply $7$ by $- 1$ .

$- \frac{1}{4} \cdot \frac{x^{7} (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4) x^{6}}{x^{14}}$

Step 3.12.2

Factor $x^{6}$ out of $x^{7} (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4) x^{6}$ .

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Step 3.12.2.1

Factor $x^{6}$ out of $x^{7} (- 8 x^{3} + 2 x^{- 3})$ .

$- \frac{1}{4} \cdot \frac{x^{6} (x (- 8 x^{3} + 2 x^{- 3})) - 7 (- 2 x^{4} - x^{- 2} - 4) x^{6}}{x^{14}}$

Step 3.12.2.2

Factor $x^{6}$ out of $- 7 (- 2 x^{4} - x^{- 2} - 4) x^{6}$ .

$- \frac{1}{4} \cdot \frac{x^{6} (x (- 8 x^{3} + 2 x^{- 3})) + x^{6} (- 7 (- 2 x^{4} - x^{- 2} - 4))}{x^{14}}$

Step 3.12.2.3

Factor $x^{6}$ out of $x^{6} (x (- 8 x^{3} + 2 x^{- 3})) + x^{6} (- 7 (- 2 x^{4} - x^{- 2} - 4))$ .

$- \frac{1}{4} \cdot \frac{x^{6} (x (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4))}{x^{14}}$

Step 4

Cancel the common factors.

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Step 4.1

Factor $x^{6}$ out of $x^{14}$ .

$- \frac{1}{4} \cdot \frac{x^{6} (x (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4))}{x^{6} x^{8}}$

Step 4.2

Cancel the common factor.

$- \frac{1}{4} \cdot \frac{x^{6} (x (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4))}{x^{6} x^{8}}$

Step 4.3

Rewrite the expression.

$- \frac{1}{4} \cdot \frac{x (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4)}{x^{8}}$

Step 5

Multiply $\frac{x (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4)}{x^{8}}$ by $\frac{1}{4}$ .

$- \frac{x (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4)}{x^{8} \cdot 4}$

Step 6

Move $4$ to the left of $x^{8}$ .

$- \frac{x (- 8 x^{3} + 2 x^{- 3}) - 7 (- 2 x^{4} - x^{- 2} - 4)}{4 \cdot x^{8}}$

Step 7

Simplify.

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Step 7.1

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

$- \frac{x (- 8 x^{3} + 2 \frac{1}{x^{3}}) - 7 (- 2 x^{4} - x^{- 2} - 4)}{4 x^{8}}$

Step 7.2

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

$- \frac{x (- 8 x^{3} + 2 \frac{1}{x^{3}}) - 7 (- 2 x^{4} - \frac{1}{x^{2}} - 4)}{4 x^{8}}$

Step 7.3

Apply the distributive property.

$- \frac{x (- 8 x^{3}) + x (2 \frac{1}{x^{3}}) - 7 (- 2 x^{4} - \frac{1}{x^{2}} - 4)}{4 x^{8}}$

Step 7.4

Apply the distributive property.

$- \frac{x (- 8 x^{3}) + x (2 \frac{1}{x^{3}}) - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Simplify each term.

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Step 7.5.1.1

Rewrite using the commutative property of multiplication.

$- \frac{- 8 x \cdot x^{3} + x (2 \frac{1}{x^{3}}) - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.2

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

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Step 7.5.1.2.1

Move $x^{3}$ .

$- \frac{- 8 (x^{3} x) + x (2 \frac{1}{x^{3}}) - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.2.2

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

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Step 7.5.1.2.2.1

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

$- \frac{- 8 (x^{3} x^{1}) + x (2 \frac{1}{x^{3}}) - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.2.2.2

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

$- \frac{- 8 x^{3 + 1} + x (2 \frac{1}{x^{3}}) - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.2.3

Add $3$ and $1$ .

$- \frac{- 8 x^{4} + x (2 \frac{1}{x^{3}}) - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.3

Rewrite using the commutative property of multiplication.

$- \frac{- 8 x^{4} + 2 x \frac{1}{x^{3}} - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.4

Cancel the common factor of $x$ .

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Step 7.5.1.4.1

Factor $x$ out of $2 x$ .

$- \frac{- 8 x^{4} + x \cdot 2 \frac{1}{x^{3}} - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.4.2

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

$- \frac{- 8 x^{4} + x \cdot 2 \frac{1}{x \cdot x^{2}} - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.4.3

Cancel the common factor.

$- \frac{- 8 x^{4} + x \cdot 2 \frac{1}{x \cdot x^{2}} - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.4.4

Rewrite the expression.

$- \frac{- 8 x^{4} + 2 \frac{1}{x^{2}} - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.5

Combine $2$ and $\frac{1}{x^{2}}$ .

$- \frac{- 8 x^{4} + \frac{2}{x^{2}} - 7 (- 2 x^{4}) - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.6

Multiply $- 2$ by $- 7$ .

$- \frac{- 8 x^{4} + \frac{2}{x^{2}} + 14 x^{4} - 7 (- \frac{1}{x^{2}}) - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.7

Multiply $- 7 (- \frac{1}{x^{2}})$ .

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Step 7.5.1.7.1

Multiply $- 1$ by $- 7$ .

$- \frac{- 8 x^{4} + \frac{2}{x^{2}} + 14 x^{4} + 7 \frac{1}{x^{2}} - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.7.2

Combine $7$ and $\frac{1}{x^{2}}$ .

$- \frac{- 8 x^{4} + \frac{2}{x^{2}} + 14 x^{4} + \frac{7}{x^{2}} - 7 \cdot - 4}{4 x^{8}}$

Step 7.5.1.8

Multiply $- 7$ by $- 4$ .

$- \frac{- 8 x^{4} + \frac{2}{x^{2}} + 14 x^{4} + \frac{7}{x^{2}} + 28}{4 x^{8}}$

Step 7.5.2

Combine the numerators over the common denominator.

$- \frac{- 8 x^{4} + 14 x^{4} + 28 + \frac{2 + 7}{x^{2}}}{4 x^{8}}$

Step 7.5.3

Add $2$ and $7$ .

$- \frac{- 8 x^{4} + 14 x^{4} + 28 + \frac{9}{x^{2}}}{4 x^{8}}$

Step 7.5.4

Add $- 8 x^{4}$ and $14 x^{4}$ .

$- \frac{6 x^{4} + 28 + \frac{9}{x^{2}}}{4 x^{8}}$

Step 7.6

Combine terms.

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Step 7.6.1

Multiply $\frac{6 x^{4} + 28 + \frac{9}{x^{2}}}{4 x^{8}}$ by $\frac{x^{2}}{x^{2}}$ .

$- (\frac{x^{2}}{x^{2}} \cdot \frac{6 x^{4} + 28 + \frac{9}{x^{2}}}{4 x^{8}})$

Step 7.6.2

Combine.

$- \frac{x^{2} (6 x^{4} + 28 + \frac{9}{x^{2}})}{x^{2} (4 x^{8})}$

Step 7.6.3

Apply the distributive property.

$- \frac{x^{2} (6 x^{4}) + x^{2} \cdot 28 + x^{2} \frac{9}{x^{2}}}{x^{2} (4 x^{8})}$

Step 7.6.4

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

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Step 7.6.4.1

Cancel the common factor.

$- \frac{x^{2} (6 x^{4}) + x^{2} \cdot 28 + x^{2} \frac{9}{x^{2}}}{x^{2} (4 x^{8})}$

Step 7.6.4.2

Rewrite the expression.

$- \frac{x^{2} (6 x^{4}) + x^{2} \cdot 28 + 9}{x^{2} (4 x^{8})}$

Step 7.6.5

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 7.6.5.1

Move $x^{4}$ .

$- \frac{x^{4} x^{2} \cdot 6 + x^{2} \cdot 28 + 9}{x^{2} (4 x^{8})}$

Step 7.6.5.2

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

$- \frac{x^{4 + 2} \cdot 6 + x^{2} \cdot 28 + 9}{x^{2} (4 x^{8})}$

Step 7.6.5.3

Add $4$ and $2$ .

$- \frac{x^{6} \cdot 6 + x^{2} \cdot 28 + 9}{x^{2} (4 x^{8})}$

Step 7.6.6

Move $6$ to the left of $x^{6}$ .

$- \frac{6 \cdot x^{6} + x^{2} \cdot 28 + 9}{x^{2} (4 x^{8})}$

Step 7.6.7

Move $28$ to the left of $x^{2}$ .

$- \frac{6 x^{6} + 28 \cdot x^{2} + 9}{x^{2} (4 x^{8})}$

Step 7.6.8

Multiply $x^{2}$ by $x^{8}$ by adding the exponents.

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Step 7.6.8.1

Move $x^{8}$ .

$- \frac{6 x^{6} + 28 x^{2} + 9}{x^{8} x^{2} \cdot 4}$

Step 7.6.8.2

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

$- \frac{6 x^{6} + 28 x^{2} + 9}{x^{8 + 2} \cdot 4}$

Step 7.6.8.3

Add $8$ and $2$ .

$- \frac{6 x^{6} + 28 x^{2} + 9}{x^{10} \cdot 4}$

Step 7.6.9

Move $4$ to the left of $x^{10}$ .

$- \frac{6 x^{6} + 28 x^{2} + 9}{4 x^{10}}$