Calculus Examples

$q (x) = (x^{- 2} - x^{- 3}) (3 x^{- 1} + 4 x^{- 4})$

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 $f (x) = x^{- 2} - x^{- 3}$ and $g (x) = 3 x^{- 1} + 4 x^{- 4}$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $3 x^{- 1} + 4 x^{- 4}$ with respect to $x$ is $\frac{d}{d x} [3 x^{- 1}] + \frac{d}{d x} [4 x^{- 4}]$ .

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

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{- 1}$ with respect to $x$ is $3 \frac{d}{d x} [x^{- 1}]$ .

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $3$ .

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

Step 2.5

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

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

Step 2.6

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

$(x^{- 2} - x^{- 3}) (- 3 x^{- 2} + 4 (- 4 x^{- 5})) + (3 x^{- 1} + 4 x^{- 4}) \frac{d}{d x} [x^{- 2} - x^{- 3}]$

Step 2.7

Multiply $- 4$ by $4$ .

$(x^{- 2} - x^{- 3}) (- 3 x^{- 2} - 16 x^{- 5}) + (3 x^{- 1} + 4 x^{- 4}) \frac{d}{d x} [x^{- 2} - x^{- 3}]$

Step 2.8

By the Sum Rule, the derivative of $x^{- 2} - x^{- 3}$ with respect to $x$ is $\frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [- x^{- 3}]$ .

$(x^{- 2} - x^{- 3}) (- 3 x^{- 2} - 16 x^{- 5}) + (3 x^{- 1} + 4 x^{- 4}) (\frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [- x^{- 3}])$

Step 2.9

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

$(x^{- 2} - x^{- 3}) (- 3 x^{- 2} - 16 x^{- 5}) + (3 x^{- 1} + 4 x^{- 4}) (- 2 x^{- 3} + \frac{d}{d x} [- x^{- 3}])$

Step 2.10

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{- 3}$ with respect to $x$ is $- \frac{d}{d x} [x^{- 3}]$ .

$(x^{- 2} - x^{- 3}) (- 3 x^{- 2} - 16 x^{- 5}) + (3 x^{- 1} + 4 x^{- 4}) (- 2 x^{- 3} - \frac{d}{d x} [x^{- 3}])$

Step 2.11

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

$(x^{- 2} - x^{- 3}) (- 3 x^{- 2} - 16 x^{- 5}) + (3 x^{- 1} + 4 x^{- 4}) (- 2 x^{- 3} - (- 3 x^{- 4}))$

Step 2.12

Multiply $- 3$ by $- 1$ .

$(x^{- 2} - x^{- 3}) (- 3 x^{- 2} - 16 x^{- 5}) + (3 x^{- 1} + 4 x^{- 4}) (- 2 x^{- 3} + 3 x^{- 4})$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Reorder terms.

$(- 3 x^{- 2} - 16 x^{- 5}) (x^{- 2} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

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

$(- 3 \frac{1}{x^{2}} - 16 x^{- 5}) (x^{- 2} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.1.2

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

$(\frac{- 3}{x^{2}} - 16 x^{- 5}) (x^{- 2} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.1.3

Move the negative in front of the fraction.

$(- \frac{3}{x^{2}} - 16 x^{- 5}) (x^{- 2} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.1.4

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

$(- \frac{3}{x^{2}} - 16 \frac{1}{x^{5}}) (x^{- 2} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.1.5

Combine $- 16$ and $\frac{1}{x^{5}}$ .

$(- \frac{3}{x^{2}} + \frac{- 16}{x^{5}}) (x^{- 2} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.1.6

Move the negative in front of the fraction.

$(- \frac{3}{x^{2}} - \frac{16}{x^{5}}) (x^{- 2} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.2

Simplify each term.

Tap for more steps...

Step 3.2.2.1

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

$(- \frac{3}{x^{2}} - \frac{16}{x^{5}}) (\frac{1}{x^{2}} - x^{- 3}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.2.2

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

$(- \frac{3}{x^{2}} - \frac{16}{x^{5}}) (\frac{1}{x^{2}} - \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.3

Expand $(- \frac{3}{x^{2}} - \frac{16}{x^{5}}) (\frac{1}{x^{2}} - \frac{1}{x^{3}})$ using the FOIL Method.

Tap for more steps...

Step 3.2.3.1

Apply the distributive property.

$- \frac{3}{x^{2}} (\frac{1}{x^{2}} - \frac{1}{x^{3}}) - \frac{16}{x^{5}} (\frac{1}{x^{2}} - \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.3.2

Apply the distributive property.

$- \frac{3}{x^{2}} \cdot \frac{1}{x^{2}} - \frac{3}{x^{2}} (- \frac{1}{x^{3}}) - \frac{16}{x^{5}} (\frac{1}{x^{2}} - \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.3.3

Apply the distributive property.

$- \frac{3}{x^{2}} \cdot \frac{1}{x^{2}} - \frac{3}{x^{2}} (- \frac{1}{x^{3}}) - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4

Simplify each term.

Tap for more steps...

Step 3.2.4.1

Multiply $- \frac{3}{x^{2}} \cdot \frac{1}{x^{2}}$ .

Tap for more steps...

Step 3.2.4.1.1

Multiply $\frac{1}{x^{2}}$ by $\frac{3}{x^{2}}$ .

$- \frac{3}{x^{2} x^{2}} - \frac{3}{x^{2}} (- \frac{1}{x^{3}}) - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.1.2

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

Tap for more steps...

Step 3.2.4.1.2.1

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

$- \frac{3}{x^{2 + 2}} - \frac{3}{x^{2}} (- \frac{1}{x^{3}}) - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.1.2.2

Add $2$ and $2$ .

$- \frac{3}{x^{4}} - \frac{3}{x^{2}} (- \frac{1}{x^{3}}) - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.2

Multiply $- \frac{3}{x^{2}} (- \frac{1}{x^{3}})$ .

Tap for more steps...

Step 3.2.4.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{3}{x^{4}} + 1 \frac{3}{x^{2}} \frac{1}{x^{3}} - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.2.2

Multiply $\frac{3}{x^{2}}$ by $1$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{2}} \cdot \frac{1}{x^{3}} - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.2.3

Multiply $\frac{3}{x^{2}}$ by $\frac{1}{x^{3}}$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{2} x^{3}} - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.2.4

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

Tap for more steps...

Step 3.2.4.2.4.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{2 + 3}} - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.2.4.2

Add $2$ and $3$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{5}} \cdot \frac{1}{x^{2}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.3

Multiply $- \frac{16}{x^{5}} \cdot \frac{1}{x^{2}}$ .

Tap for more steps...

Step 3.2.4.3.1

Multiply $\frac{1}{x^{2}}$ by $\frac{16}{x^{5}}$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{2} x^{5}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.3.2

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

Tap for more steps...

Step 3.2.4.3.2.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{2 + 5}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.3.2.2

Add $2$ and $5$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} - \frac{16}{x^{5}} (- \frac{1}{x^{3}}) + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.4

Multiply $- \frac{16}{x^{5}} (- \frac{1}{x^{3}})$ .

Tap for more steps...

Step 3.2.4.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + 1 \frac{16}{x^{5}} \frac{1}{x^{3}} + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.4.2

Multiply $\frac{16}{x^{5}}$ by $1$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{5}} \cdot \frac{1}{x^{3}} + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.4.3

Multiply $\frac{16}{x^{5}}$ by $\frac{1}{x^{3}}$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{5} x^{3}} + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.4.4

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

Tap for more steps...

Step 3.2.4.4.4.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{5 + 3}} + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.4.4.4.2

Add $5$ and $3$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- 2 x^{- 3} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.5

Simplify each term.

Tap for more steps...

Step 3.2.5.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- 2 \frac{1}{x^{3}} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.5.2

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (\frac{- 2}{x^{3}} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.5.3

Move the negative in front of the fraction.

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- \frac{2}{x^{3}} + 3 x^{- 4}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.5.4

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- \frac{2}{x^{3}} + 3 \frac{1}{x^{4}}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.5.5

Combine $3$ and $\frac{1}{x^{4}}$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- \frac{2}{x^{3}} + \frac{3}{x^{4}}) (3 x^{- 1} + 4 x^{- 4})$

Step 3.2.6

Simplify each term.

Tap for more steps...

Step 3.2.6.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- \frac{2}{x^{3}} + \frac{3}{x^{4}}) (3 \frac{1}{x} + 4 x^{- 4})$

Step 3.2.6.2

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- \frac{2}{x^{3}} + \frac{3}{x^{4}}) (\frac{3}{x} + 4 x^{- 4})$

Step 3.2.6.3

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- \frac{2}{x^{3}} + \frac{3}{x^{4}}) (\frac{3}{x} + 4 \frac{1}{x^{4}})$

Step 3.2.6.4

Combine $4$ and $\frac{1}{x^{4}}$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} + (- \frac{2}{x^{3}} + \frac{3}{x^{4}}) (\frac{3}{x} + \frac{4}{x^{4}})$

Step 3.2.7

Expand $(- \frac{2}{x^{3}} + \frac{3}{x^{4}}) (\frac{3}{x} + \frac{4}{x^{4}})$ using the FOIL Method.

Tap for more steps...

Step 3.2.7.1

Apply the distributive property.

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{2}{x^{3}} (\frac{3}{x} + \frac{4}{x^{4}}) + \frac{3}{x^{4}} (\frac{3}{x} + \frac{4}{x^{4}})$

Step 3.2.7.2

Apply the distributive property.

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{2}{x^{3}} \cdot \frac{3}{x} - \frac{2}{x^{3}} \cdot \frac{4}{x^{4}} + \frac{3}{x^{4}} (\frac{3}{x} + \frac{4}{x^{4}})$

Step 3.2.7.3

Apply the distributive property.

Step 3.2.8

Simplify each term.

Tap for more steps...

Step 3.2.8.1

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

Tap for more steps...

Step 3.2.8.1.1

Multiply $\frac{3}{x}$ by $\frac{2}{x^{3}}$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{3 \cdot 2}{x \cdot x^{3}} - \frac{2}{x^{3}} \cdot \frac{4}{x^{4}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.1.2

Multiply $3$ by $2$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x \cdot x^{3}} - \frac{2}{x^{3}} \cdot \frac{4}{x^{4}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.1.3

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

Tap for more steps...

Step 3.2.8.1.3.1

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

Tap for more steps...

Step 3.2.8.1.3.1.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{1} x^{3}} - \frac{2}{x^{3}} \cdot \frac{4}{x^{4}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.1.3.1.2

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{1 + 3}} - \frac{2}{x^{3}} \cdot \frac{4}{x^{4}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.1.3.2

Add $1$ and $3$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{2}{x^{3}} \cdot \frac{4}{x^{4}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.2

Multiply $- \frac{2}{x^{3}} \cdot \frac{4}{x^{4}}$ .

Tap for more steps...

Step 3.2.8.2.1

Multiply $\frac{4}{x^{4}}$ by $\frac{2}{x^{3}}$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{4 \cdot 2}{x^{4} x^{3}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.2.2

Multiply $4$ by $2$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{4} x^{3}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.2.3

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

Tap for more steps...

Step 3.2.8.2.3.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{4 + 3}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.2.3.2

Add $4$ and $3$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{3}{x^{4}} \cdot \frac{3}{x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.3

Combine.

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{3 \cdot 3}{x^{4} x} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.4

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

Tap for more steps...

Step 3.2.8.4.1

Multiply $x^{4}$ by $x$ .

Tap for more steps...

Step 3.2.8.4.1.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{3 \cdot 3}{x^{4} x^{1}} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.4.1.2

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{3 \cdot 3}{x^{4 + 1}} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.4.2

Add $4$ and $1$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{3 \cdot 3}{x^{5}} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.5

Multiply $3$ by $3$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{9}{x^{5}} + \frac{3}{x^{4}} \cdot \frac{4}{x^{4}}$

Step 3.2.8.6

Combine.

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{9}{x^{5}} + \frac{3 \cdot 4}{x^{4} x^{4}}$

Step 3.2.8.7

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

Tap for more steps...

Step 3.2.8.7.1

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

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{9}{x^{5}} + \frac{3 \cdot 4}{x^{4 + 4}}$

Step 3.2.8.7.2

Add $4$ and $4$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{9}{x^{5}} + \frac{3 \cdot 4}{x^{8}}$

Step 3.2.8.8

Multiply $3$ by $4$ .

$- \frac{3}{x^{4}} + \frac{3}{x^{5}} - \frac{16}{x^{7}} + \frac{16}{x^{8}} - \frac{6}{x^{4}} - \frac{8}{x^{7}} + \frac{9}{x^{5}} + \frac{12}{x^{8}}$

Step 3.3

Combine the numerators over the common denominator.

$\frac{- 3 - 6}{x^{4}} + \frac{3 + 9}{x^{5}} + \frac{- 16 - 8}{x^{7}} + \frac{16 + 12}{x^{8}}$

Step 3.4

Subtract $6$ from $- 3$ .

$\frac{- 9}{x^{4}} + \frac{3 + 9}{x^{5}} + \frac{- 16 - 8}{x^{7}} + \frac{16 + 12}{x^{8}}$

Step 3.5

Add $3$ and $9$ .

$\frac{- 9}{x^{4}} + \frac{12}{x^{5}} + \frac{- 16 - 8}{x^{7}} + \frac{16 + 12}{x^{8}}$

Step 3.6

Subtract $8$ from $- 16$ .

$\frac{- 9}{x^{4}} + \frac{12}{x^{5}} + \frac{- 24}{x^{7}} + \frac{16 + 12}{x^{8}}$

Step 3.7

Add $16$ and $12$ .

$\frac{- 9}{x^{4}} + \frac{12}{x^{5}} + \frac{- 24}{x^{7}} + \frac{28}{x^{8}}$

Step 3.8

Simplify each term.

Tap for more steps...

Step 3.8.1

Move the negative in front of the fraction.

$- \frac{9}{x^{4}} + \frac{12}{x^{5}} + \frac{- 24}{x^{7}} + \frac{28}{x^{8}}$

Step 3.8.2

Move the negative in front of the fraction.

$- \frac{9}{x^{4}} + \frac{12}{x^{5}} - \frac{24}{x^{7}} + \frac{28}{x^{8}}$