Calculus Examples

$y = (x^{0.3} - 3 x - 6) (x^{- 1} + x^{- 2})$

Step 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 $f (x) = x^{0.3} - 3 x - 6$ and $g (x) = x^{- 1} + x^{- 2}$ .

$(x^{0.3} - 3 x - 6) \frac{d}{d x} [x^{- 1} + x^{- 2}] + (x^{- 1} + x^{- 2}) \frac{d}{d x} [x^{0.3} - 3 x - 6]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{- 1} + x^{- 2}$ with respect to $x$ is $\frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [x^{- 2}]$ .

$(x^{0.3} - 3 x - 6) (\frac{d}{d x} [x^{- 1}] + \frac{d}{d x} [x^{- 2}]) + (x^{- 1} + x^{- 2}) \frac{d}{d x} [x^{0.3} - 3 x - 6]$

Step 2.2

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

$(x^{0.3} - 3 x - 6) (- x^{- 2} + \frac{d}{d x} [x^{- 2}]) + (x^{- 1} + x^{- 2}) \frac{d}{d x} [x^{0.3} - 3 x - 6]$

Step 2.3

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

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) \frac{d}{d x} [x^{0.3} - 3 x - 6]$

Step 2.4

By the Sum Rule, the derivative of $x^{0.3} - 3 x - 6$ with respect to $x$ is $\frac{d}{d x} [x^{0.3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 6]$ .

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (\frac{d}{d x} [x^{0.3}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 6])$

Step 2.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.3$ .

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 6])$

Step 2.6

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

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 6])$

Step 2.7

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

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} - 3 \cdot 1 + \frac{d}{d x} [- 6])$

Step 2.8

Multiply $- 3$ by $1$ .

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} - 3 + \frac{d}{d x} [- 6])$

Step 2.9

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

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} - 3 + 0)$

Step 2.10

Add $0.3 x^{- 0.7} - 3$ and $0$ .

$(x^{0.3} - 3 x - 6) (- x^{- 2} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} - 3)$

Step 3

Simplify.

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Step 3.1

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

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} - 2 x^{- 3}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} - 3)$

Step 3.2

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

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} - 2 \frac{1}{x^{3}}) + (x^{- 1} + x^{- 2}) (0.3 x^{- 0.7} - 3)$

Step 3.3

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

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} - 2 \frac{1}{x^{3}}) + (\frac{1}{x} + x^{- 2}) (0.3 x^{- 0.7} - 3)$

Step 3.4

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

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} - 2 \frac{1}{x^{3}}) + (\frac{1}{x} + \frac{1}{x^{2}}) (0.3 x^{- 0.7} - 3)$

Step 3.5

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

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} - 2 \frac{1}{x^{3}}) + (\frac{1}{x} + \frac{1}{x^{2}}) (0.3 \frac{1}{x^{0.7}} - 3)$

Step 3.6

Combine terms.

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Step 3.6.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} + \frac{- 2}{x^{3}}) + (\frac{1}{x} + \frac{1}{x^{2}}) (0.3 \frac{1}{x^{0.7}} - 3)$

Step 3.6.2

Move the negative in front of the fraction.

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} - \frac{2}{x^{3}}) + (\frac{1}{x} + \frac{1}{x^{2}}) (0.3 \frac{1}{x^{0.7}} - 3)$

Step 3.6.3

Combine $0.3$ and $\frac{1}{x^{0.7}}$ .

$(x^{0.3} - 3 x - 6) (- \frac{1}{x^{2}} - \frac{2}{x^{3}}) + (\frac{1}{x} + \frac{1}{x^{2}}) (\frac{0.3}{x^{0.7}} - 3)$

Step 3.7

Reorder terms.

$(- \frac{1}{x^{2}} - \frac{2}{x^{3}}) (x^{0.3} - 3 x - 6) + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8

Simplify each term.

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Step 3.8.1

Expand $(- \frac{1}{x^{2}} - \frac{2}{x^{3}}) (x^{0.3} - 3 x - 6)$ by multiplying each term in the first expression by each term in the second expression.

$- \frac{1}{x^{2}} x^{0.3} - \frac{1}{x^{2}} (- 3 x) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

Cancel the common factor of $x^{0.3}$ .

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Step 3.8.2.1.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$\frac{- 1}{x^{2}} x^{0.3} - \frac{1}{x^{2}} (- 3 x) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.1.2

Factor $x^{0.3}$ out of $x^{2}$ .

$\frac{- 1}{x^{0.3} x^{1.7}} x^{0.3} - \frac{1}{x^{2}} (- 3 x) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.1.3

Cancel the common factor.

Step 3.8.2.1.4

Rewrite the expression.

$\frac{- 1}{x^{1.7}} - \frac{1}{x^{2}} (- 3 x) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.2

Move the negative in front of the fraction.

$- \frac{1}{x^{1.7}} - \frac{1}{x^{2}} (- 3 x) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.3

Cancel the common factor of $x$ .

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Step 3.8.2.3.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$- \frac{1}{x^{1.7}} + \frac{- 1}{x^{2}} (- 3 x) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.3.2

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

$- \frac{1}{x^{1.7}} + \frac{- 1}{x \cdot x} (- 3 x) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.3.3

Factor $x$ out of $- 3 x$ .

$- \frac{1}{x^{1.7}} + \frac{- 1}{x \cdot x} (x \cdot - 3) - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.3.4

Cancel the common factor.

Step 3.8.2.3.5

Rewrite the expression.

$- \frac{1}{x^{1.7}} + \frac{- 1}{x} \cdot - 3 - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.4

Combine $\frac{- 1}{x}$ and $- 3$ .

$- \frac{1}{x^{1.7}} + \frac{- - 3}{x} - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.5

Multiply $- 1$ by $- 3$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} - \frac{1}{x^{2}} \cdot - 6 - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.6

Multiply $- \frac{1}{x^{2}} \cdot - 6$ .

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Step 3.8.2.6.1

Multiply $- 6$ by $- 1$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + 6 \frac{1}{x^{2}} - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.6.2

Combine $6$ and $\frac{1}{x^{2}}$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.7

Cancel the common factor of $x^{0.3}$ .

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Step 3.8.2.7.1

Move the leading negative in $- \frac{2}{x^{3}}$ into the numerator.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} + \frac{- 2}{x^{3}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.7.2

Factor $x^{0.3}$ out of $x^{3}$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} + \frac{- 2}{x^{0.3} x^{2.7}} x^{0.3} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.7.3

Cancel the common factor.

Step 3.8.2.7.4

Rewrite the expression.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} + \frac{- 2}{x^{2.7}} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.8

Move the negative in front of the fraction.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} - \frac{2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.9

Cancel the common factor of $x$ .

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Step 3.8.2.9.1

Move the leading negative in $- \frac{2}{x^{3}}$ into the numerator.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{- 2}{x^{3}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.9.2

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

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{- 2}{x \cdot x^{2}} (- 3 x) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.9.3

Factor $x$ out of $- 3 x$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{- 2}{x \cdot x^{2}} (x \cdot - 3) - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.9.4

Cancel the common factor.

Step 3.8.2.9.5

Rewrite the expression.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{- 2}{x^{2}} \cdot - 3 - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.10

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

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{- 2 \cdot - 3}{x^{2}} - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.11

Multiply $- 2$ by $- 3$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{6}{x^{2}} - \frac{2}{x^{3}} \cdot - 6 + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.12

Multiply $- \frac{2}{x^{3}} \cdot - 6$ .

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Step 3.8.2.12.1

Multiply $- 6$ by $- 1$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{6}{x^{2}} + 6 \frac{2}{x^{3}} + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.12.2

Combine $6$ and $\frac{2}{x^{3}}$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{6}{x^{2}} + \frac{6 \cdot 2}{x^{3}} + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.2.12.3

Multiply $6$ by $2$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{6}{x^{2}} - \frac{2}{x^{2.7}} + \frac{6}{x^{2}} + \frac{12}{x^{3}} + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.3

Combine the numerators over the common denominator.

$\frac{- 1}{x^{1.7}} + \frac{3}{x} + \frac{6 + 6}{x^{2}} + \frac{- 2}{x^{2.7}} + \frac{12}{x^{3}} + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.4

Add $6$ and $6$ .

$\frac{- 1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} + \frac{- 2}{x^{2.7}} + \frac{12}{x^{3}} + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.5

Simplify each term.

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Step 3.8.5.1

Move the negative in front of the fraction.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} + \frac{- 2}{x^{2.7}} + \frac{12}{x^{3}} + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.5.2

Move the negative in front of the fraction.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + (\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.6

Expand $(\frac{0.3}{x^{0.7}} - 3) (\frac{1}{x} + \frac{1}{x^{2}})$ using the FOIL Method.

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Step 3.8.6.1

Apply the distributive property.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{0.7}} (\frac{1}{x} + \frac{1}{x^{2}}) - 3 (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.6.2

Apply the distributive property.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{0.7}} \cdot \frac{1}{x} + \frac{0.3}{x^{0.7}} \cdot \frac{1}{x^{2}} - 3 (\frac{1}{x} + \frac{1}{x^{2}})$

Step 3.8.6.3

Apply the distributive property.

Step 3.8.7

Simplify each term.

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Step 3.8.7.1

Combine.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3 \cdot 1}{x^{0.7} x} + \frac{0.3}{x^{0.7}} \cdot \frac{1}{x^{2}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.2

Multiply $x^{0.7}$ by $x$ by adding the exponents.

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Step 3.8.7.2.1

Multiply $x^{0.7}$ by $x$ .

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Step 3.8.7.2.1.1

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

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3 \cdot 1}{x^{0.7} x^{1}} + \frac{0.3}{x^{0.7}} \cdot \frac{1}{x^{2}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.2.1.2

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

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3 \cdot 1}{x^{0.7 + 1}} + \frac{0.3}{x^{0.7}} \cdot \frac{1}{x^{2}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.2.2

Add $0.7$ and $1$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3 \cdot 1}{x^{1.7}} + \frac{0.3}{x^{0.7}} \cdot \frac{1}{x^{2}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.3

Multiply $0.3$ by $1$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{0.7}} \cdot \frac{1}{x^{2}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.4

Combine.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3 \cdot 1}{x^{0.7} x^{2}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.5

Multiply $x^{0.7}$ by $x^{2}$ by adding the exponents.

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Step 3.8.7.5.1

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

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3 \cdot 1}{x^{0.7 + 2}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.5.2

Add $0.7$ and $2$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3 \cdot 1}{x^{2.7}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.6

Multiply $0.3$ by $1$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} - 3 \frac{1}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.7

Combine $- 3$ and $\frac{1}{x}$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} + \frac{- 3}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.8

Move the negative in front of the fraction.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} - \frac{3}{x} - 3 \frac{1}{x^{2}}$

Step 3.8.7.9

Combine $- 3$ and $\frac{1}{x^{2}}$ .

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} - \frac{3}{x} + \frac{- 3}{x^{2}}$

Step 3.8.7.10

Move the negative in front of the fraction.

$- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} - \frac{3}{x} - \frac{3}{x^{2}}$

Step 3.9

Combine the opposite terms in $- \frac{1}{x^{1.7}} + \frac{3}{x} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} - \frac{3}{x} - \frac{3}{x^{2}}$ .

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Step 3.9.1

Subtract $\frac{3}{x}$ from $\frac{3}{x}$ .

$- \frac{1}{x^{1.7}} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} + 0 - \frac{3}{x^{2}}$

Step 3.9.2

Add $- \frac{1}{x^{1.7}} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}}$ and $0$ .

$- \frac{1}{x^{1.7}} + \frac{12}{x^{2}} - \frac{2}{x^{2.7}} + \frac{12}{x^{3}} + \frac{0.3}{x^{1.7}} + \frac{0.3}{x^{2.7}} - \frac{3}{x^{2}}$

Step 3.10

Combine the numerators over the common denominator.

$\frac{- 1 + 0.3}{x^{1.7}} + \frac{12 - 3}{x^{2}} + \frac{- 2 + 0.3}{x^{2.7}} + \frac{12}{x^{3}}$

Step 3.11

Add $- 1$ and $0.3$ .

$\frac{- 0.7}{x^{1.7}} + \frac{12 - 3}{x^{2}} + \frac{- 2 + 0.3}{x^{2.7}} + \frac{12}{x^{3}}$

Step 3.12

Subtract $3$ from $12$ .

$\frac{- 0.7}{x^{1.7}} + \frac{9}{x^{2}} + \frac{- 2 + 0.3}{x^{2.7}} + \frac{12}{x^{3}}$

Step 3.13

Add $- 2$ and $0.3$ .

$\frac{- 0.7}{x^{1.7}} + \frac{9}{x^{2}} + \frac{- 1.7}{x^{2.7}} + \frac{12}{x^{3}}$

Step 3.14

Simplify each term.

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Step 3.14.1

Move the negative in front of the fraction.

$- \frac{0.7}{x^{1.7}} + \frac{9}{x^{2}} + \frac{- 1.7}{x^{2.7}} + \frac{12}{x^{3}}$

Step 3.14.2

Move the negative in front of the fraction.

$- \frac{0.7}{x^{1.7}} + \frac{9}{x^{2}} - \frac{1.7}{x^{2.7}} + \frac{12}{x^{3}}$