Algebra Examples

$y = {(6 x - 5)}^{2} {(3 - x^{5})}^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} ({(6 x - 5)}^{2} {(3 - x^{5})}^{2})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Rewrite ${(6 x - 5)}^{2}$ as $(6 x - 5) (6 x - 5)$ .

$\frac{d}{d y} [(6 x - 5) (6 x - 5) {(3 - x^{5})}^{2}]$

Step 3.2

Expand $(6 x - 5) (6 x - 5)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{d}{d y} [(6 x (6 x - 5) - 5 (6 x - 5)) {(3 - x^{5})}^{2}]$

Step 3.2.2

Apply the distributive property.

$\frac{d}{d y} [(6 x (6 x) + 6 x \cdot - 5 - 5 (6 x - 5)) {(3 - x^{5})}^{2}]$

Step 3.2.3

Apply the distributive property.

$\frac{d}{d y} [(6 x (6 x) + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5) {(3 - x^{5})}^{2}]$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d y} [(6 \cdot 6 x \cdot x + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5) {(3 - x^{5})}^{2}]$

Step 3.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.2.1

Move $x$ .

$\frac{d}{d y} [(6 \cdot 6 (x \cdot x) + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5) {(3 - x^{5})}^{2}]$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d y} [(6 \cdot 6 x^{2} + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5) {(3 - x^{5})}^{2}]$

Step 3.3.1.3

Multiply $6$ by $6$ .

$\frac{d}{d y} [(36 x^{2} + 6 x \cdot - 5 - 5 (6 x) - 5 \cdot - 5) {(3 - x^{5})}^{2}]$

Step 3.3.1.4

Multiply $- 5$ by $6$ .

$\frac{d}{d y} [(36 x^{2} - 30 x - 5 (6 x) - 5 \cdot - 5) {(3 - x^{5})}^{2}]$

Step 3.3.1.5

Multiply $6$ by $- 5$ .

$\frac{d}{d y} [(36 x^{2} - 30 x - 30 x - 5 \cdot - 5) {(3 - x^{5})}^{2}]$

Step 3.3.1.6

Multiply $- 5$ by $- 5$ .

$\frac{d}{d y} [(36 x^{2} - 30 x - 30 x + 25) {(3 - x^{5})}^{2}]$

Step 3.3.2

Subtract $30 x$ from $- 30 x$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) {(3 - x^{5})}^{2}]$

Step 3.4

Rewrite ${(3 - x^{5})}^{2}$ as $(3 - x^{5}) (3 - x^{5})$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) ((3 - x^{5}) (3 - x^{5}))]$

Step 3.5

Expand $(3 - x^{5}) (3 - x^{5})$ using the FOIL Method.

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Step 3.5.1

Apply the distributive property.

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (3 (3 - x^{5}) - x^{5} (3 - x^{5}))]$

Step 3.5.2

Apply the distributive property.

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (3 \cdot 3 + 3 (- x^{5}) - x^{5} (3 - x^{5}))]$

Step 3.5.3

Apply the distributive property.

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (3 \cdot 3 + 3 (- x^{5}) - x^{5} \cdot 3 - x^{5} (- x^{5}))]$

Step 3.6

Simplify and combine like terms.

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Step 3.6.1

Simplify each term.

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Step 3.6.1.1

Multiply $3$ by $3$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 + 3 (- x^{5}) - x^{5} \cdot 3 - x^{5} (- x^{5}))]$

Step 3.6.1.2

Multiply $- 1$ by $3$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - x^{5} \cdot 3 - x^{5} (- x^{5}))]$

Step 3.6.1.3

Multiply $3$ by $- 1$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - 3 x^{5} - x^{5} (- x^{5}))]$

Step 3.6.1.4

Rewrite using the commutative property of multiplication.

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - 3 x^{5} - 1 \cdot - 1 x^{5} x^{5})]$

Step 3.6.1.5

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

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Step 3.6.1.5.1

Move $x^{5}$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - 3 x^{5} - 1 \cdot - 1 (x^{5} x^{5}))]$

Step 3.6.1.5.2

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

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - 3 x^{5} - 1 \cdot - 1 x^{5 + 5})]$

Step 3.6.1.5.3

Add $5$ and $5$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - 3 x^{5} - 1 \cdot - 1 x^{10})]$

Step 3.6.1.6

Multiply $- 1$ by $- 1$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - 3 x^{5} + 1 x^{10})]$

Step 3.6.1.7

Multiply $x^{10}$ by $1$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 3 x^{5} - 3 x^{5} + x^{10})]$

Step 3.6.2

Subtract $3 x^{5}$ from $- 3 x^{5}$ .

$\frac{d}{d y} [(36 x^{2} - 60 x + 25) (9 - 6 x^{5} + x^{10})]$

Step 3.7

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 $f (y) = 36 x^{2} - 60 x + 25$ and $g (y) = 9 - 6 x^{5} + x^{10}$ .

$(36 x^{2} - 60 x + 25) \frac{d}{d y} [9 - 6 x^{5} + x^{10}] + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.8

Differentiate.

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Step 3.8.1

By the Sum Rule, the derivative of $9 - 6 x^{5} + x^{10}$ with respect to $y$ is $\frac{d}{d y} [9] + \frac{d}{d y} [- 6 x^{5}] + \frac{d}{d y} [x^{10}]$ .

$(36 x^{2} - 60 x + 25) (\frac{d}{d y} [9] + \frac{d}{d y} [- 6 x^{5}] + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.8.2

Since $9$ is constant with respect to $y$ , the derivative of $9$ with respect to $y$ is $0$ .

$(36 x^{2} - 60 x + 25) (0 + \frac{d}{d y} [- 6 x^{5}] + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.8.3

Add $0$ and $\frac{d}{d y} [- 6 x^{5}]$ .

$(36 x^{2} - 60 x + 25) (\frac{d}{d y} [- 6 x^{5}] + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.8.4

Since $- 6$ is constant with respect to $y$ , the derivative of $- 6 x^{5}$ with respect to $y$ is $- 6 \frac{d}{d y} [x^{5}]$ .

$(36 x^{2} - 60 x + 25) (- 6 \frac{d}{d y} [x^{5}] + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.9

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{5}$ and $g (y) = x$ .

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Step 3.9.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

$(36 x^{2} - 60 x + 25) (- 6 (\frac{d}{d u_{1}} [{u_{1}}^{5}] \frac{d}{d y} [x]) + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.9.2

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

$(36 x^{2} - 60 x + 25) (- 6 (5 {u_{1}}^{4} \frac{d}{d y} [x]) + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.9.3

Replace all occurrences of $u_{1}$ with $x$ .

$(36 x^{2} - 60 x + 25) (- 6 (5 x^{4} \frac{d}{d y} [x]) + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.10

Multiply $5$ by $- 6$ .

$(36 x^{2} - 60 x + 25) (- 30 (x^{4} \frac{d}{d y} [x]) + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.11

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + \frac{d}{d y} [x^{10}]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.12

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{10}$ and $g (y) = x$ .

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Step 3.12.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + \frac{d}{d u_{2}} [{u_{2}}^{10}] \frac{d}{d y} [x]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.12.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 10$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 {u_{2}}^{9} \frac{d}{d y} [x]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.12.3

Replace all occurrences of $u_{2}$ with $x$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} \frac{d}{d y} [x]) + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.13

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) \frac{d}{d y} [36 x^{2} - 60 x + 25]$

Step 3.14

Differentiate.

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Step 3.14.1

By the Sum Rule, the derivative of $36 x^{2} - 60 x + 25$ with respect to $y$ is $\frac{d}{d y} [36 x^{2}] + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25]$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (\frac{d}{d y} [36 x^{2}] + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25])$

Step 3.14.2

Since $36$ is constant with respect to $y$ , the derivative of $36 x^{2}$ with respect to $y$ is $36 \frac{d}{d y} [x^{2}]$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (36 \frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25])$

Step 3.15

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 3.15.1

To apply the Chain Rule, set $u_{3}$ as $x$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (36 (\frac{d}{d u_{3}} [{u_{3}}^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25])$

Step 3.15.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = 2$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (36 (2 u_{3} \frac{d}{d y} [x]) + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25])$

Step 3.15.3

Replace all occurrences of $u_{3}$ with $x$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (36 (2 x \frac{d}{d y} [x]) + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25])$

Step 3.16

Multiply $2$ by $36$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 (x \frac{d}{d y} [x]) + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25])$

Step 3.17

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' + \frac{d}{d y} [- 60 x] + \frac{d}{d y} [25])$

Step 3.18

Since $- 60$ is constant with respect to $y$ , the derivative of $- 60 x$ with respect to $y$ is $- 60 \frac{d}{d y} [x]$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' - 60 \frac{d}{d y} [x] + \frac{d}{d y} [25])$

Step 3.19

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' - 60 x' + \frac{d}{d y} [25])$

Step 3.20

Since $25$ is constant with respect to $y$ , the derivative of $25$ with respect to $y$ is $0$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' - 60 x' + 0)$

Step 3.21

Add $72 x x' - 60 x'$ and $0$ .

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x' + 10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' - 60 x')$

Step 3.22

Simplify.

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Step 3.22.1

Apply the distributive property.

$(36 x^{2} - 60 x + 25) (- 30 x^{4} x') + (36 x^{2} - 60 x + 25) (10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' - 60 x')$

Step 3.22.2

Apply the distributive property.

$36 x^{2} (- 30 x^{4} x') - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + (36 x^{2} - 60 x + 25) (10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' - 60 x')$

Step 3.22.3

Apply the distributive property.

$36 x^{2} (- 30 x^{4} x') - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x' - 60 x')$

Step 3.22.4

Apply the distributive property.

$36 x^{2} (- 30 x^{4} x') - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + (9 - 6 x^{5} + x^{10}) (72 x x') + (9 - 6 x^{5} + x^{10}) (- 60 x')$

Step 3.22.5

Apply the distributive property.

$36 x^{2} (- 30 x^{4} x') - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + (9 - 6 x^{5} + x^{10}) (- 60 x')$

Step 3.22.6

Apply the distributive property.

$36 x^{2} (- 30 x^{4} x') - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7

Combine terms.

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Step 3.22.7.1

Multiply $- 30$ by $36$ .

$- 1080 x^{2} (x^{4} x') - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.2

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

$- 1080 x^{4 + 2} x' - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.3

Add $4$ and $2$ .

$- 1080 x^{6} x' - 60 x (- 30 x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.4

Multiply $- 30$ by $- 60$ .

$- 1080 x^{6} x' + 1800 x (x^{4} x') + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.5

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

$- 1080 x^{6} x' + 1800 (x^{4} x^{1}) x' + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.6

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

$- 1080 x^{6} x' + 1800 x^{4 + 1} x' + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.7

Add $4$ and $1$ .

$- 1080 x^{6} x' + 1800 x^{5} x' + 25 (- 30 x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.8

Multiply $- 30$ by $25$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 (x^{4} x') + 36 x^{2} (10 x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.9

Multiply $10$ by $36$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{2} (x^{9} x') - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.10

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

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{9 + 2} x' - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.11

Add $9$ and $2$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 60 x (10 x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.12

Multiply $10$ by $- 60$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x (x^{9} x') + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.13

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

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 (x^{9} x^{1}) x' + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.14

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

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{9 + 1} x' + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.15

Add $9$ and $1$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 25 (10 x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.16

Multiply $10$ by $25$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 (x^{9} x') + 9 (72 x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.17

Multiply $72$ by $9$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 (x x') - 6 x^{5} (72 x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.18

Multiply $72$ by $- 6$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{5} (x x') + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.19

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

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 (x^{1} x^{5}) x' + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.20

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

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{1 + 5} x' + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.21

Add $1$ and $5$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{6} x' + x^{10} (72 x x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.22

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

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Step 3.22.7.22.1

Move $x$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{6} x' + x \cdot x^{10} (72 x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.22.2

Multiply $x$ by $x^{10}$ .

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Step 3.22.7.22.2.1

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

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{6} x' + x^{1} x^{10} (72 x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.22.2.2

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

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{6} x' + x^{1 + 10} (72 x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.22.3

Add $1$ and $10$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{6} x' + x^{11} (72 x') + 9 (- 60 x') - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.23

Multiply $- 60$ by $9$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{6} x' + x^{11} (72 x') - 540 x' - 6 x^{5} (- 60 x') + x^{10} (- 60 x')$

Step 3.22.7.24

Multiply $- 60$ by $- 6$ .

$- 1080 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 432 x^{6} x' + x^{11} (72 x') - 540 x' + 360 x^{5} x' + x^{10} (- 60 x')$

Step 3.22.7.25

Subtract $432 x^{6} x'$ from $- 1080 x^{6} x'$ .

$- 1512 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' + x^{11} (72 x') - 540 x' + 360 x^{5} x' + x^{10} (- 60 x')$

Step 3.22.7.26

Add $360 x^{11} x'$ and $x^{11} (72 x')$ .

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Step 3.22.7.26.1

Reorder $x^{11}$ and $72$ .

$- 1512 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 360 x^{11} x' + 72 \cdot x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' + 360 x^{5} x' + x^{10} (- 60 x')$

Step 3.22.7.26.2

Add $360 x^{11} x'$ and $72 \cdot x^{11} x'$ .

$- 1512 x^{6} x' + 1800 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' + 360 x^{5} x' + x^{10} (- 60 x')$

Step 3.22.7.27

Add $1800 x^{5} x'$ and $360 x^{5} x'$ .

$- 1512 x^{6} x' + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 600 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' + x^{10} (- 60 x')$

Step 3.22.7.28

Add $- 600 x^{10} x'$ and $x^{10} (- 60 x')$ .

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Step 3.22.7.28.1

Reorder $x^{10}$ and $- 60$ .

$- 1512 x^{6} x' + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 600 x^{10} x' - 60 \cdot x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x'$

Step 3.22.7.28.2

Subtract $60 \cdot x^{10} x'$ from $- 600 x^{10} x'$ .

$- 1512 x^{6} x' + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x'$

Step 4

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

$1 = - 1512 x^{6} x' + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $- 1512 x^{6} x' + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' = 1$ .

$- 1512 x^{6} x' + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' = 1$

Step 5.2

Factor $2 x'$ out of $- 1512 x^{6} x' + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x'$ .

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Step 5.2.1

Factor $2 x'$ out of $- 1512 x^{6} x'$ .

$2 x' (- 756 x^{6}) + 2160 x^{5} x' - 750 x^{4} x' + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' = 1$

Step 5.2.2

Factor $2 x'$ out of $2160 x^{5} x'$ .

$2 x' (- 756 x^{6}) + 2 x' (1080 x^{5}) - 750 x^{4} x' + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' = 1$

Step 5.2.3

Factor $2 x'$ out of $- 750 x^{4} x'$ .

$2 x' (- 756 x^{6}) + 2 x' (1080 x^{5}) + 2 x' (- 375 x^{4}) + 432 x^{11} x' - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' = 1$

Step 5.2.4

Factor $2 x'$ out of $432 x^{11} x'$ .

$2 x' (- 756 x^{6}) + 2 x' (1080 x^{5}) + 2 x' (- 375 x^{4}) + 2 x' (216 x^{11}) - 660 x^{10} x' + 250 x^{9} x' + 648 x x' - 540 x' = 1$

Step 5.2.5

Factor $2 x'$ out of $- 660 x^{10} x'$ .

$2 x' (- 756 x^{6}) + 2 x' (1080 x^{5}) + 2 x' (- 375 x^{4}) + 2 x' (216 x^{11}) + 2 x' (- 330 x^{10}) + 250 x^{9} x' + 648 x x' - 540 x' = 1$

Step 5.2.6

Factor $2 x'$ out of $250 x^{9} x'$ .

$2 x' (- 756 x^{6}) + 2 x' (1080 x^{5}) + 2 x' (- 375 x^{4}) + 2 x' (216 x^{11}) + 2 x' (- 330 x^{10}) + 2 x' (125 x^{9}) + 648 x x' - 540 x' = 1$

Step 5.2.7

Factor $2 x'$ out of $648 x x'$ .

$2 x' (- 756 x^{6}) + 2 x' (1080 x^{5}) + 2 x' (- 375 x^{4}) + 2 x' (216 x^{11}) + 2 x' (- 330 x^{10}) + 2 x' (125 x^{9}) + 2 x' (324 x) - 540 x' = 1$

Step 5.2.8

Factor $2 x'$ out of $- 540 x'$ .

$2 x' (- 756 x^{6}) + 2 x' (1080 x^{5}) + 2 x' (- 375 x^{4}) + 2 x' (216 x^{11}) + 2 x' (- 330 x^{10}) + 2 x' (125 x^{9}) + 2 x' (324 x) + 2 x' \cdot - 270 = 1$

Step 5.2.9

Factor $2 x'$ out of $2 x' (- 756 x^{6}) + 2 x' (1080 x^{5})$ .

$2 x' (- 756 x^{6} + 1080 x^{5}) + 2 x' (- 375 x^{4}) + 2 x' (216 x^{11}) + 2 x' (- 330 x^{10}) + 2 x' (125 x^{9}) + 2 x' (324 x) + 2 x' \cdot - 270 = 1$

Step 5.2.10

Factor $2 x'$ out of $2 x' (- 756 x^{6} + 1080 x^{5}) + 2 x' (- 375 x^{4})$ .

$2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4}) + 2 x' (216 x^{11}) + 2 x' (- 330 x^{10}) + 2 x' (125 x^{9}) + 2 x' (324 x) + 2 x' \cdot - 270 = 1$

Step 5.2.11

Factor $2 x'$ out of $2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4}) + 2 x' (216 x^{11})$ .

$2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11}) + 2 x' (- 330 x^{10}) + 2 x' (125 x^{9}) + 2 x' (324 x) + 2 x' \cdot - 270 = 1$

Step 5.2.12

Factor $2 x'$ out of $2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11}) + 2 x' (- 330 x^{10})$ .

$2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11} - 330 x^{10}) + 2 x' (125 x^{9}) + 2 x' (324 x) + 2 x' \cdot - 270 = 1$

Step 5.2.13

Factor $2 x'$ out of $2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11} - 330 x^{10}) + 2 x' (125 x^{9})$ .

$2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11} - 330 x^{10} + 125 x^{9}) + 2 x' (324 x) + 2 x' \cdot - 270 = 1$

Step 5.2.14

Factor $2 x'$ out of $2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11} - 330 x^{10} + 125 x^{9}) + 2 x' (324 x)$ .

$2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11} - 330 x^{10} + 125 x^{9} + 324 x) + 2 x' \cdot - 270 = 1$

Step 5.2.15

Factor $2 x'$ out of $2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11} - 330 x^{10} + 125 x^{9} + 324 x) + 2 x' \cdot - 270$ .

$2 x' (- 756 x^{6} + 1080 x^{5} - 375 x^{4} + 216 x^{11} - 330 x^{10} + 125 x^{9} + 324 x - 270) = 1$

Step 5.3

Reorder terms.

$2 x' (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270) = 1$

Step 5.4

Divide each term in $2 x' (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270) = 1$ by $2 (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)$ and simplify.

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Step 5.4.1

$\frac{2 x' (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)}{2 (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)} = \frac{1}{2 (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x' (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)}{216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270} = \frac{1}{2 (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)}$

Step 5.4.2.2

Cancel the common factor of $216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270$ .

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Step 5.4.2.2.1

Cancel the common factor.

Step 5.4.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{1}{2 (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{2 (216 x^{11} - 330 x^{10} + 125 x^{9} - 756 x^{6} + 1080 x^{5} - 375 x^{4} + 324 x - 270)}$