Calculus Examples

${(2 x + 2 y)}^{3} = 8 x^{3} + 8 y^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(2 x + 2 y)}^{3}) = \frac{d}{d x} (8 x^{3} + 8 y^{3})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 2 x + 2 y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $2 x + 2 y$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [2 x + 2 y]$

Step 2.1.2

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

$3 u^{2} \frac{d}{d x} [2 x + 2 y]$

Step 2.1.3

Replace all occurrences of $u$ with $2 x + 2 y$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $2$ by $1$ .

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

Step 2.2.5

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

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

Step 2.3

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

$3 {(2 x + 2 y)}^{2} (2 + 2 y')$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$3 {(2 x + 2 y)}^{2} \cdot 2 + 3 {(2 x + 2 y)}^{2} (2 y')$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $2$ by $3$ .

$6 {(2 x + 2 y)}^{2} + 3 {(2 x + 2 y)}^{2} (2 y')$

Step 2.4.2.2

Multiply $2$ by $3$ .

$6 {(2 x + 2 y)}^{2} + 6 {(2 x + 2 y)}^{2} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [8 y^{3}]$

Step 3.2

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

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Step 3.2.1

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

$8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [8 y^{3}]$

Step 3.2.2

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

$8 (3 x^{2}) + \frac{d}{d x} [8 y^{3}]$

Step 3.2.3

Multiply $3$ by $8$ .

$24 x^{2} + \frac{d}{d x} [8 y^{3}]$

Step 3.3

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

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Step 3.3.1

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

$24 x^{2} + 8 \frac{d}{d x} [y^{3}]$

Step 3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $y$ .

$24 x^{2} + 8 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [y])$

Step 3.3.2.2

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

$24 x^{2} + 8 (3 u^{2} \frac{d}{d x} [y])$

Step 3.3.2.3

Replace all occurrences of $u$ with $y$ .

$24 x^{2} + 8 (3 y^{2} \frac{d}{d x} [y])$

Step 3.3.3

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

$24 x^{2} + 8 (3 y^{2} y')$

Step 3.3.4

Multiply $3$ by $8$ .

$24 x^{2} + 24 y^{2} y'$

Step 4

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

$6 {(2 x + 2 y)}^{2} + 6 {(2 x + 2 y)}^{2} y' = 24 x^{2} + 24 y^{2} y'$

Step 5

Solve for $y'$ .

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Step 5.1

Reorder factors in $6 {(2 x + 2 y)}^{2} + 6 {(2 x + 2 y)}^{2} y'$ .

$6 {(2 x + 2 y)}^{2} + 6 y' {(2 x + 2 y)}^{2} = 24 x^{2} + 24 y^{2} y'$

Step 5.2

Move all terms containing $y'$ to the left side of the equation.

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Step 5.2.1

Subtract $24 y^{2} y'$ from both sides of the equation.

$6 {(2 x + 2 y)}^{2} + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Rewrite ${(2 x + 2 y)}^{2}$ as $(2 x + 2 y) (2 x + 2 y)$ .

$6 ((2 x + 2 y) (2 x + 2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.2

Expand $(2 x + 2 y) (2 x + 2 y)$ using the FOIL Method.

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Step 5.2.2.2.1

Apply the distributive property.

$6 (2 x (2 x + 2 y) + 2 y (2 x + 2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.2.2

Apply the distributive property.

$6 (2 x (2 x) + 2 x (2 y) + 2 y (2 x + 2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.2.3

Apply the distributive property.

$6 (2 x (2 x) + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3

Simplify and combine like terms.

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Step 5.2.2.3.1

Simplify each term.

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Step 5.2.2.3.1.1

Rewrite using the commutative property of multiplication.

$6 (2 \cdot 2 x \cdot x + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.2

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

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Step 5.2.2.3.1.2.1

Move $x$ .

$6 (2 \cdot 2 (x \cdot x) + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.2.2

Multiply $x$ by $x$ .

$6 (2 \cdot 2 x^{2} + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.3

Multiply $2$ by $2$ .

$6 (4 x^{2} + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.4

Rewrite using the commutative property of multiplication.

$6 (4 x^{2} + 2 \cdot 2 x y + 2 y (2 x) + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.5

Multiply $2$ by $2$ .

$6 (4 x^{2} + 4 x y + 2 y (2 x) + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.6

Rewrite using the commutative property of multiplication.

$6 (4 x^{2} + 4 x y + 2 \cdot 2 y x + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.7

Multiply $2$ by $2$ .

$6 (4 x^{2} + 4 x y + 4 y x + 2 y (2 y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.8

Rewrite using the commutative property of multiplication.

$6 (4 x^{2} + 4 x y + 4 y x + 2 \cdot 2 y \cdot y) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.9

Multiply $y$ by $y$ by adding the exponents.

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Step 5.2.2.3.1.9.1

Move $y$ .

$6 (4 x^{2} + 4 x y + 4 y x + 2 \cdot 2 (y \cdot y)) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.9.2

Multiply $y$ by $y$ .

$6 (4 x^{2} + 4 x y + 4 y x + 2 \cdot 2 y^{2}) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.1.10

Multiply $2$ by $2$ .

$6 (4 x^{2} + 4 x y + 4 y x + 4 y^{2}) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.2

Add $4 x y$ and $4 y x$ .

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Step 5.2.2.3.2.1

Move $y$ .

$6 (4 x^{2} + 4 x y + 4 x y + 4 y^{2}) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.3.2.2

Add $4 x y$ and $4 x y$ .

$6 (4 x^{2} + 8 x y + 4 y^{2}) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.4

Apply the distributive property.

$6 (4 x^{2}) + 6 (8 x y) + 6 (4 y^{2}) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.5

Simplify.

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Step 5.2.2.5.1

Multiply $4$ by $6$ .

$24 x^{2} + 6 (8 x y) + 6 (4 y^{2}) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.5.2

Multiply $8$ by $6$ .

$24 x^{2} + 48 (x y) + 6 (4 y^{2}) + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.5.3

Multiply $4$ by $6$ .

$24 x^{2} + 48 (x y) + 24 y^{2} + 6 y' {(2 x + 2 y)}^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.6

Rewrite ${(2 x + 2 y)}^{2}$ as $(2 x + 2 y) (2 x + 2 y)$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' ((2 x + 2 y) (2 x + 2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.7

Expand $(2 x + 2 y) (2 x + 2 y)$ using the FOIL Method.

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Step 5.2.2.7.1

Apply the distributive property.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (2 x (2 x + 2 y) + 2 y (2 x + 2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.7.2

Apply the distributive property.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (2 x (2 x) + 2 x (2 y) + 2 y (2 x + 2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.7.3

Apply the distributive property.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (2 x (2 x) + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8

Simplify and combine like terms.

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Step 5.2.2.8.1

Simplify each term.

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Step 5.2.2.8.1.1

Rewrite using the commutative property of multiplication.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (2 \cdot 2 x \cdot x + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.2

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

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Step 5.2.2.8.1.2.1

Move $x$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (2 \cdot 2 (x \cdot x) + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.2.2

Multiply $x$ by $x$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (2 \cdot 2 x^{2} + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.3

Multiply $2$ by $2$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 2 x (2 y) + 2 y (2 x) + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.4

Rewrite using the commutative property of multiplication.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 2 \cdot 2 x y + 2 y (2 x) + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.5

Multiply $2$ by $2$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 2 y (2 x) + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.6

Rewrite using the commutative property of multiplication.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 2 \cdot 2 y x + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.7

Multiply $2$ by $2$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 4 y x + 2 y (2 y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.8

Rewrite using the commutative property of multiplication.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 4 y x + 2 \cdot 2 y \cdot y) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.9

Multiply $y$ by $y$ by adding the exponents.

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Step 5.2.2.8.1.9.1

Move $y$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 4 y x + 2 \cdot 2 (y \cdot y)) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.9.2

Multiply $y$ by $y$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 4 y x + 2 \cdot 2 y^{2}) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.1.10

Multiply $2$ by $2$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 4 y x + 4 y^{2}) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.2

Add $4 x y$ and $4 y x$ .

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Step 5.2.2.8.2.1

Move $y$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 4 x y + 4 x y + 4 y^{2}) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.8.2.2

Add $4 x y$ and $4 x y$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2} + 8 x y + 4 y^{2}) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.9

Apply the distributive property.

$24 x^{2} + 48 x y + 24 y^{2} + 6 y' (4 x^{2}) + 6 y' (8 x y) + 6 y' (4 y^{2}) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.10

Simplify.

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Step 5.2.2.10.1

Rewrite using the commutative property of multiplication.

$24 x^{2} + 48 x y + 24 y^{2} + 6 \cdot 4 y' x^{2} + 6 y' (8 x y) + 6 y' (4 y^{2}) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.10.2

Multiply $8$ by $6$ .

$24 x^{2} + 48 x y + 24 y^{2} + 6 \cdot 4 y' x^{2} + 48 y' (x y) + 6 y' (4 y^{2}) - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.10.3

Rewrite using the commutative property of multiplication.

$24 x^{2} + 48 x y + 24 y^{2} + 6 \cdot 4 y' x^{2} + 48 y' (x y) + 6 \cdot 4 y' y^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.11

Simplify each term.

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Step 5.2.2.11.1

Multiply $6$ by $4$ .

$24 x^{2} + 48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' (x y) + 6 \cdot 4 y' y^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.2.11.2

Multiply $6$ by $4$ .

$24 x^{2} + 48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' x y + 24 y' y^{2} - 24 y^{2} y' = 24 x^{2}$

Step 5.2.3

Combine the opposite terms in $24 x^{2} + 48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' x y + 24 y' y^{2} - 24 y^{2} y'$ .

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Step 5.2.3.1

Reorder the factors in the terms $24 y' y^{2}$ and $- 24 y^{2} y'$ .

$24 x^{2} + 48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' x y + 24 y^{2} y' - 24 y^{2} y' = 24 x^{2}$

Step 5.2.3.2

Subtract $24 y^{2} y'$ from $24 y^{2} y'$ .

$24 x^{2} + 48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' x y + 0 = 24 x^{2}$

Step 5.2.3.3

Add $24 x^{2} + 48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' x y$ and $0$ .

$24 x^{2} + 48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' x y = 24 x^{2}$

Step 5.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.3.1

Subtract $24 x^{2}$ from both sides of the equation.

$48 x y + 24 y^{2} + 24 y' x^{2} + 48 y' x y = 24 x^{2} - 24 x^{2}$

Step 5.3.2

Subtract $48 x y$ from both sides of the equation.

$24 y^{2} + 24 y' x^{2} + 48 y' x y = 24 x^{2} - 24 x^{2} - 48 x y$

Step 5.3.3

Subtract $24 y^{2}$ from both sides of the equation.

$24 y' x^{2} + 48 y' x y = 24 x^{2} - 24 x^{2} - 48 x y - 24 y^{2}$

Step 5.3.4

Subtract $24 x^{2}$ from $24 x^{2}$ .

$24 y' x^{2} + 48 y' x y = 0 - 48 x y - 24 y^{2}$

Step 5.3.5

Subtract $48 x y$ from $0$ .

$24 y' x^{2} + 48 y' x y = - 48 x y - 24 y^{2}$

Step 5.4

Factor $24 y' x$ out of $24 y' x^{2} + 48 y' x y$ .

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Step 5.4.1

Factor $24 y' x$ out of $24 y' x^{2}$ .

$24 y' x \cdot x + 48 y' x y = - 48 x y - 24 y^{2}$

Step 5.4.2

Factor $24 y' x$ out of $48 y' x y$ .

$24 y' x \cdot x + 24 y' x (2 y) = - 48 x y - 24 y^{2}$

Step 5.4.3

Factor $24 y' x$ out of $24 y' x \cdot x + 24 y' x (2 y)$ .

$24 y' x (x + 2 y) = - 48 x y - 24 y^{2}$

Step 5.5

Divide each term in $24 y' x (x + 2 y) = - 48 x y - 24 y^{2}$ by $24 x (x + 2 y)$ and simplify.

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Step 5.5.1

Divide each term in $24 y' x (x + 2 y) = - 48 x y - 24 y^{2}$ by $24 x (x + 2 y)$ .

$\frac{24 y' x (x + 2 y)}{24 x (x + 2 y)} = \frac{- 48 x y}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $24$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{24 y' x (x + 2 y)}{24 x (x + 2 y)} = \frac{- 48 x y}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y' x (x + 2 y)}{x (x + 2 y)} = \frac{- 48 x y}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.2.2

Cancel the common factor of $x$ .

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{y' x (x + 2 y)}{x (x + 2 y)} = \frac{- 48 x y}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.2.2.2

Rewrite the expression.

$\frac{y' (x + 2 y)}{x + 2 y} = \frac{- 48 x y}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.2.3

Cancel the common factor of $x + 2 y$ .

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Step 5.5.2.3.1

Cancel the common factor.

$\frac{y' (x + 2 y)}{x + 2 y} = \frac{- 48 x y}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- 48 x y}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Cancel the common factor of $- 48$ and $24$ .

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Step 5.5.3.1.1.1

Factor $24$ out of $- 48 x y$ .

$y' = \frac{24 (- 2 x y)}{24 x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

Factor $24$ out of $24 x (x + 2 y)$ .

$y' = \frac{24 (- 2 x y)}{24 (x (x + 2 y))} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{24 (- 2 x y)}{24 (x (x + 2 y))} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- 2 x y}{x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3.1.2

Cancel the common factor of $x$ .

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Step 5.5.3.1.2.1

Cancel the common factor.

$y' = \frac{- 2 x y}{x (x + 2 y)} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3.1.2.2

Rewrite the expression.

$y' = \frac{- 2 y}{x + 2 y} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3.1.3

Move the negative in front of the fraction.

$y' = - \frac{2 y}{x + 2 y} + \frac{- 24 y^{2}}{24 x (x + 2 y)}$

Step 5.5.3.1.4

Cancel the common factor of $- 24$ and $24$ .

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Step 5.5.3.1.4.1

Factor $24$ out of $- 24 y^{2}$ .

$y' = - \frac{2 y}{x + 2 y} + \frac{24 (- y^{2})}{24 x (x + 2 y)}$

Step 5.5.3.1.4.2

Cancel the common factors.

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Step 5.5.3.1.4.2.1

Factor $24$ out of $24 x (x + 2 y)$ .

$y' = - \frac{2 y}{x + 2 y} + \frac{24 (- y^{2})}{24 (x (x + 2 y))}$

Step 5.5.3.1.4.2.2

Cancel the common factor.

$y' = - \frac{2 y}{x + 2 y} + \frac{24 (- y^{2})}{24 (x (x + 2 y))}$

Step 5.5.3.1.4.2.3

Rewrite the expression.

$y' = - \frac{2 y}{x + 2 y} + \frac{- y^{2}}{x (x + 2 y)}$

Step 5.5.3.1.5

Move the negative in front of the fraction.

$y' = - \frac{2 y}{x + 2 y} - \frac{y^{2}}{x (x + 2 y)}$

Step 5.5.3.2

To write $- \frac{2 y}{x + 2 y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = - \frac{2 y}{x + 2 y} \cdot \frac{x}{x} - \frac{y^{2}}{x (x + 2 y)}$

Step 5.5.3.3

Write each expression with a common denominator of $(x + 2 y) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.1

Multiply $\frac{2 y}{x + 2 y}$ by $\frac{x}{x}$ .

$y' = - \frac{2 y x}{(x + 2 y) x} - \frac{y^{2}}{x (x + 2 y)}$

Step 5.5.3.3.2

Reorder the factors of $(x + 2 y) x$ .

$y' = - \frac{2 y x}{x (x + 2 y)} - \frac{y^{2}}{x (x + 2 y)}$

Step 5.5.3.4

Combine the numerators over the common denominator.

$y' = \frac{- 2 y x - y^{2}}{x (x + 2 y)}$

Step 5.5.3.5

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

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Step 5.5.3.5.1

Factor $y$ out of $- 2 y x$ .

$y' = \frac{y (- 2 x) - y^{2}}{x (x + 2 y)}$

Step 5.5.3.5.2

Factor $y$ out of $- y^{2}$ .

$y' = \frac{y (- 2 x) + y (- y)}{x (x + 2 y)}$

Step 5.5.3.5.3

Factor $y$ out of $y (- 2 x) + y (- y)$ .

$y' = \frac{y (- 2 x - y)}{x (x + 2 y)}$

Step 5.5.3.6

Factor $- 1$ out of $- 2 x$ .

$y' = \frac{y (- (2 x) - y)}{x (x + 2 y)}$

Step 5.5.3.7

Factor $- 1$ out of $- y$ .

$y' = \frac{y (- (2 x) - (y))}{x (x + 2 y)}$

Step 5.5.3.8

Factor $- 1$ out of $- (2 x) - (y)$ .

$y' = \frac{y (- (2 x + y))}{x (x + 2 y)}$

Step 5.5.3.9

Simplify the expression.

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Step 5.5.3.9.1

Rewrite $- (2 x + y)$ as $- 1 (2 x + y)$ .

$y' = \frac{y (- 1 (2 x + y))}{x (x + 2 y)}$

Step 5.5.3.9.2

Move the negative in front of the fraction.

$y' = - \frac{y (2 x + y)}{x (x + 2 y)}$

Step 6

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

$\frac{d y}{d x} = - \frac{y (2 x + y)}{x (x + 2 y)}$