Find the Derivative Using Product Rule - d/dy (y^-1+y^-2)(8y^-3-9y^-4)

$(y^{- 1} + y^{- 2}) (8 y^{- 3} - 9 y^{- 4})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y^{- 1} + y^{- 2}$ and $g (y) = 8 y^{- 3} - 9 y^{- 4}$ .

$(y^{- 1} + y^{- 2}) \frac{d}{d y} [8 y^{- 3} - 9 y^{- 4}] + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 2

By the Sum Rule, the derivative of $8 y^{- 3} - 9 y^{- 4}$ with respect to $y$ is $\frac{d}{d y} [8 y^{- 3}] + \frac{d}{d y} [- 9 y^{- 4}]$ .

$(y^{- 1} + y^{- 2}) (\frac{d}{d y} [8 y^{- 3}] + \frac{d}{d y} [- 9 y^{- 4}]) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 3

Evaluate $\frac{d}{d y} [8 y^{- 3}]$ .

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

$(y^{- 1} + y^{- 2}) (8 \frac{d}{d y} [y^{- 3}] + \frac{d}{d y} [- 9 y^{- 4}]) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

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

$(y^{- 1} + y^{- 2}) (8 (- 3 y^{- 4}) + \frac{d}{d y} [- 9 y^{- 4}]) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Multiply $- 3$ by $8$ .

$(y^{- 1} + y^{- 2}) (- 24 y^{- 4} + \frac{d}{d y} [- 9 y^{- 4}]) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 4

Evaluate $\frac{d}{d y} [- 9 y^{- 4}]$ .

Since $- 9$ is constant with respect to $y$ , the derivative of $- 9 y^{- 4}$ with respect to $y$ is $- 9 \frac{d}{d y} [y^{- 4}]$ .

$(y^{- 1} + y^{- 2}) (- 24 y^{- 4} - 9 \frac{d}{d y} [y^{- 4}]) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

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

$(y^{- 1} + y^{- 2}) (- 24 y^{- 4} - 9 (- 4 y^{- 5})) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Multiply $- 4$ by $- 9$ .

$(y^{- 1} + y^{- 2}) (- 24 y^{- 4} + 36 y^{- 5}) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Step 5

Differentiate.

Simplify each term.

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

$(y^{- 1} + y^{- 2}) (- 24 \frac{1}{y^{4}} + 36 y^{- 5}) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Combine $- 24$ and $\frac{1}{y^{4}}$ .

$(y^{- 1} + y^{- 2}) (\frac{- 24}{y^{4}} + 36 y^{- 5}) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Move the negative in front of the fraction.

$(y^{- 1} + y^{- 2}) (- \frac{24}{y^{4}} + 36 y^{- 5}) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

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

$(y^{- 1} + y^{- 2}) (- \frac{24}{y^{4}} + 36 \frac{1}{y^{5}}) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

Combine $36$ and $\frac{1}{y^{5}}$ .

$(y^{- 1} + y^{- 2}) (- \frac{24}{y^{4}} + \frac{36}{y^{5}}) + (8 y^{- 3} - 9 y^{- 4}) \frac{d}{d y} [y^{- 1} + y^{- 2}]$

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

$(y^{- 1} + y^{- 2}) (- \frac{24}{y^{4}} + \frac{36}{y^{5}}) + (8 y^{- 3} - 9 y^{- 4}) (\frac{d}{d y} [y^{- 1}] + \frac{d}{d y} [y^{- 2}])$

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

$(y^{- 1} + y^{- 2}) (- \frac{24}{y^{4}} + \frac{36}{y^{5}}) + (8 y^{- 3} - 9 y^{- 4}) (- y^{- 2} + \frac{d}{d y} [y^{- 2}])$

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

$(y^{- 1} + y^{- 2}) (- \frac{24}{y^{4}} + \frac{36}{y^{5}}) + (8 y^{- 3} - 9 y^{- 4}) (- y^{- 2} - 2 y^{- 3})$

Step 6

Simplify.

Reorder terms.

$(- \frac{24}{y^{4}} + \frac{36}{y^{5}}) (y^{- 1} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Simplify each term.

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

$(- \frac{24}{y^{4}} + \frac{36}{y^{5}}) (\frac{1}{y} + y^{- 2}) + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

$(- \frac{24}{y^{4}} + \frac{36}{y^{5}}) (\frac{1}{y} + \frac{1}{y^{2}}) + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Expand $(- \frac{24}{y^{4}} + \frac{36}{y^{5}}) (\frac{1}{y} + \frac{1}{y^{2}})$ using the FOIL Method.

Apply the distributive property.

$- \frac{24}{y^{4}} (\frac{1}{y} + \frac{1}{y^{2}}) + \frac{36}{y^{5}} (\frac{1}{y} + \frac{1}{y^{2}}) + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Apply the distributive property.

$- \frac{24}{y^{4}} \cdot \frac{1}{y} - \frac{24}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} (\frac{1}{y} + \frac{1}{y^{2}}) + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Apply the distributive property.

$- \frac{24}{y^{4}} \cdot \frac{1}{y} - \frac{24}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Simplify and combine like terms.

Simplify each term.

Multiply $- \frac{24}{y^{4}} \cdot \frac{1}{y}$ .

Multiply $\frac{1}{y}$ by $\frac{24}{y^{4}}$ .

$- \frac{24}{y \cdot y^{4}} - \frac{24}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

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

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

$- \frac{24}{y^{1} y^{4}} - \frac{24}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

$- \frac{24}{y^{1 + 4}} - \frac{24}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Add $1$ and $4$ .

$- \frac{24}{y^{5}} - \frac{24}{y^{4}} \cdot \frac{1}{y^{2}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Multiply $- \frac{24}{y^{4}} \cdot \frac{1}{y^{2}}$ .

Multiply $\frac{1}{y^{2}}$ by $\frac{24}{y^{4}}$ .

$- \frac{24}{y^{5}} - \frac{24}{y^{2} y^{4}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

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

$- \frac{24}{y^{5}} - \frac{24}{y^{2 + 4}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Add $2$ and $4$ .

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36}{y^{5}} \cdot \frac{1}{y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Combine.

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36 \cdot 1}{y^{5} y} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

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

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

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36 \cdot 1}{y^{5} y^{1}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36 \cdot 1}{y^{5 + 1}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Add $5$ and $1$ .

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36 \cdot 1}{y^{6}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Multiply $36$ by $1$ .

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36}{y^{6}} + \frac{36}{y^{5}} \cdot \frac{1}{y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Combine.

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36}{y^{6}} + \frac{36 \cdot 1}{y^{5} y^{2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

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

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36}{y^{6}} + \frac{36 \cdot 1}{y^{5 + 2}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Add $5$ and $2$ .

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36}{y^{6}} + \frac{36 \cdot 1}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Multiply $36$ by $1$ .

$- \frac{24}{y^{5}} - \frac{24}{y^{6}} + \frac{36}{y^{6}} + \frac{36}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Combine the numerators over the common denominator.

$- \frac{24}{y^{5}} + \frac{- 24 + 36}{y^{6}} + \frac{36}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Add $- 24$ and $36$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- y^{- 2} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

Simplify each term.

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - 2 y^{- 3}) (8 y^{- 3} - 9 y^{- 4})$

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - 2 \frac{1}{y^{3}}) (8 y^{- 3} - 9 y^{- 4})$

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} + \frac{- 2}{y^{3}}) (8 y^{- 3} - 9 y^{- 4})$

Move the negative in front of the fraction.

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (8 y^{- 3} - 9 y^{- 4})$

Simplify each term.

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (8 \frac{1}{y^{3}} - 9 y^{- 4})$

Combine $8$ and $\frac{1}{y^{3}}$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{8}{y^{3}} - 9 y^{- 4})$

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{8}{y^{3}} - 9 \frac{1}{y^{4}})$

Combine $- 9$ and $\frac{1}{y^{4}}$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{8}{y^{3}} + \frac{- 9}{y^{4}})$

Move the negative in front of the fraction.

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} + (- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{8}{y^{3}} - \frac{9}{y^{4}})$

Expand $(- \frac{1}{y^{2}} - \frac{2}{y^{3}}) (\frac{8}{y^{3}} - \frac{9}{y^{4}})$ using the FOIL Method.

Apply the distributive property.

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{1}{y^{2}} (\frac{8}{y^{3}} - \frac{9}{y^{4}}) - \frac{2}{y^{3}} (\frac{8}{y^{3}} - \frac{9}{y^{4}})$

Apply the distributive property.

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{1}{y^{2}} \cdot \frac{8}{y^{3}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} (\frac{8}{y^{3}} - \frac{9}{y^{4}})$

Apply the distributive property.

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{1}{y^{2}} \cdot \frac{8}{y^{3}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Simplify and combine like terms.

Simplify each term.

Multiply $- \frac{1}{y^{2}} \cdot \frac{8}{y^{3}}$ .

Multiply $\frac{8}{y^{3}}$ by $\frac{1}{y^{2}}$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{3} y^{2}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

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

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{3 + 2}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Add $3$ and $2$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} - \frac{1}{y^{2}} (- \frac{9}{y^{4}}) - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Multiply $- \frac{1}{y^{2}} (- \frac{9}{y^{4}})$ .

Multiply $- 1$ by $- 1$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + 1 \frac{1}{y^{2}} \frac{9}{y^{4}} - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Multiply $\frac{1}{y^{2}}$ by $1$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{1}{y^{2}} \cdot \frac{9}{y^{4}} - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Multiply $\frac{1}{y^{2}}$ by $\frac{9}{y^{4}}$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{2} y^{4}} - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

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

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{2 + 4}} - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Add $2$ and $4$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{2}{y^{3}} \cdot \frac{8}{y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Multiply $- \frac{2}{y^{3}} \cdot \frac{8}{y^{3}}$ .

Multiply $\frac{8}{y^{3}}$ by $\frac{2}{y^{3}}$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{8 \cdot 2}{y^{3} y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Multiply $8$ by $2$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{3} y^{3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

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

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{3 + 3}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Add $3$ and $3$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{6}} - \frac{2}{y^{3}} (- \frac{9}{y^{4}})$

Multiply $- \frac{2}{y^{3}} (- \frac{9}{y^{4}})$ .

Multiply $- 1$ by $- 1$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{6}} + 1 \frac{2}{y^{3}} \frac{9}{y^{4}}$

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{6}} + \frac{2}{y^{3}} \cdot \frac{9}{y^{4}}$

Multiply $\frac{2}{y^{3}}$ by $\frac{9}{y^{4}}$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{6}} + \frac{2 \cdot 9}{y^{3} y^{4}}$

Multiply $2$ by $9$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{6}} + \frac{18}{y^{3} y^{4}}$

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

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

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{6}} + \frac{18}{y^{3 + 4}}$

Add $3$ and $4$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9}{y^{6}} - \frac{16}{y^{6}} + \frac{18}{y^{7}}$

Combine the numerators over the common denominator.

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{9 - 16}{y^{6}} + \frac{18}{y^{7}}$

Subtract $16$ from $9$ .

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} + \frac{- 7}{y^{6}} + \frac{18}{y^{7}}$

Move the negative in front of the fraction.

$- \frac{24}{y^{5}} + \frac{12}{y^{6}} + \frac{36}{y^{7}} - \frac{8}{y^{5}} - \frac{7}{y^{6}} + \frac{18}{y^{7}}$

Combine the numerators over the common denominator.

$\frac{- 24 - 8}{y^{5}} + \frac{12 - 7}{y^{6}} + \frac{36 + 18}{y^{7}}$

Subtract $8$ from $- 24$ .

$\frac{- 32}{y^{5}} + \frac{12 - 7}{y^{6}} + \frac{36 + 18}{y^{7}}$

Subtract $7$ from $12$ .

$\frac{- 32}{y^{5}} + \frac{5}{y^{6}} + \frac{36 + 18}{y^{7}}$

Add $36$ and $18$ .

$\frac{- 32}{y^{5}} + \frac{5}{y^{6}} + \frac{54}{y^{7}}$

Move the negative in front of the fraction.

$- \frac{32}{y^{5}} + \frac{5}{y^{6}} + \frac{54}{y^{7}}$

Find the Derivative Using Product Rule - d/dy (y^-1+y^-2)(8y^-3-9y^-4)

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