Calculus Examples

${(4 x + y)}^{4} = 3 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(4 x + y)}^{4}) = \frac{d}{d x} (3 y)$

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 $f (x) = x^{4}$ and $g (x) = 4 x + y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $4 x + y$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [4 x + y]$

Step 2.1.2

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

$4 u^{3} \frac{d}{d x} [4 x + y]$

Step 2.1.3

Replace all occurrences of $u$ with $4 x + y$ .

$4 {(4 x + y)}^{3} \frac{d}{d x} [4 x + y]$

Step 2.2

Differentiate.

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Step 2.2.1

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

$4 {(4 x + y)}^{3} (\frac{d}{d x} [4 x] + \frac{d}{d x} [y])$

Step 2.2.2

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

$4 {(4 x + y)}^{3} (4 \frac{d}{d x} [x] + \frac{d}{d x} [y])$

Step 2.2.3

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

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

Step 2.2.4

Multiply $4$ by $1$ .

$4 {(4 x + y)}^{3} (4 + \frac{d}{d x} [y])$

Step 2.3

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

$4 {(4 x + y)}^{3} (4 + y')$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$3 \frac{d}{d x} [y]$

Step 3.2

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

$3 y'$

Step 4

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

$4 {(4 x + y)}^{3} (4 + y') = 3 y'$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify $4 {(4 x + y)}^{3} (4 + y')$ .

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Step 5.1.1

Rewrite.

$0 + 0 + 4 {(4 x + y)}^{3} (4 + y') = 3 y'$

Step 5.1.2

Simplify by adding zeros.

$4 {(4 x + y)}^{3} (4 + y') = 3 y'$

Step 5.1.3

Apply the distributive property.

$4 {(4 x + y)}^{3} \cdot 4 + 4 {(4 x + y)}^{3} y' = 3 y'$

Step 5.1.4

Simplify the expression.

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Step 5.1.4.1

Multiply $4$ by $4$ .

$16 {(4 x + y)}^{3} + 4 {(4 x + y)}^{3} y' = 3 y'$

Step 5.1.4.2

Reorder factors in $16 {(4 x + y)}^{3} + 4 {(4 x + y)}^{3} y'$ .

$16 {(4 x + y)}^{3} + 4 y' {(4 x + y)}^{3} = 3 y'$

Step 5.2

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

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Step 5.2.1

Subtract $3 y'$ from both sides of the equation.

$16 {(4 x + y)}^{3} + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Use the Binomial Theorem.

$16 ({(4 x)}^{3} + 3 {(4 x)}^{2} y + 3 (4 x) y^{2} + y^{3}) + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.2

Simplify each term.

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Step 5.2.2.2.1

Apply the product rule to $4 x$ .

$16 (4^{3} x^{3} + 3 {(4 x)}^{2} y + 3 (4 x) y^{2} + y^{3}) + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.2.2

Raise $4$ to the power of $3$ .

$16 (64 x^{3} + 3 {(4 x)}^{2} y + 3 (4 x) y^{2} + y^{3}) + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.2.3

Apply the product rule to $4 x$ .

$16 (64 x^{3} + 3 (4^{2} x^{2}) y + 3 (4 x) y^{2} + y^{3}) + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.2.4

Raise $4$ to the power of $2$ .

$16 (64 x^{3} + 3 (16 x^{2}) y + 3 (4 x) y^{2} + y^{3}) + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.2.5

Multiply $16$ by $3$ .

$16 (64 x^{3} + 48 x^{2} y + 3 (4 x) y^{2} + y^{3}) + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.2.6

Multiply $4$ by $3$ .

$16 (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3}) + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.3

Apply the distributive property.

$16 (64 x^{3}) + 16 (48 x^{2} y) + 16 (12 x y^{2}) + 16 y^{3} + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.4

Simplify.

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Step 5.2.2.4.1

Multiply $64$ by $16$ .

$1024 x^{3} + 16 (48 x^{2} y) + 16 (12 x y^{2}) + 16 y^{3} + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.4.2

Multiply $48$ by $16$ .

$1024 x^{3} + 768 (x^{2} y) + 16 (12 x y^{2}) + 16 y^{3} + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.4.3

Multiply $12$ by $16$ .

$1024 x^{3} + 768 (x^{2} y) + 192 (x y^{2}) + 16 y^{3} + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.5

Remove parentheses.

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' {(4 x + y)}^{3} - 3 y' = 0$

Step 5.2.2.6

Use the Binomial Theorem.

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' ({(4 x)}^{3} + 3 {(4 x)}^{2} y + 3 (4 x) y^{2} + y^{3}) - 3 y' = 0$

Step 5.2.2.7

Simplify each term.

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Step 5.2.2.7.1

Apply the product rule to $4 x$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' (4^{3} x^{3} + 3 {(4 x)}^{2} y + 3 (4 x) y^{2} + y^{3}) - 3 y' = 0$

Step 5.2.2.7.2

Raise $4$ to the power of $3$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' (64 x^{3} + 3 {(4 x)}^{2} y + 3 (4 x) y^{2} + y^{3}) - 3 y' = 0$

Step 5.2.2.7.3

Apply the product rule to $4 x$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' (64 x^{3} + 3 (4^{2} x^{2}) y + 3 (4 x) y^{2} + y^{3}) - 3 y' = 0$

Step 5.2.2.7.4

Raise $4$ to the power of $2$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' (64 x^{3} + 3 (16 x^{2}) y + 3 (4 x) y^{2} + y^{3}) - 3 y' = 0$

Step 5.2.2.7.5

Multiply $16$ by $3$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' (64 x^{3} + 48 x^{2} y + 3 (4 x) y^{2} + y^{3}) - 3 y' = 0$

Step 5.2.2.7.6

Multiply $4$ by $3$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3}) - 3 y' = 0$

Step 5.2.2.8

Apply the distributive property.

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 y' (64 x^{3}) + 4 y' (48 x^{2} y) + 4 y' (12 x y^{2}) + 4 y' y^{3} - 3 y' = 0$

Step 5.2.2.9

Simplify.

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Step 5.2.2.9.1

Rewrite using the commutative property of multiplication.

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 \cdot 64 y' x^{3} + 4 y' (48 x^{2} y) + 4 y' (12 x y^{2}) + 4 y' y^{3} - 3 y' = 0$

Step 5.2.2.9.2

Multiply $48$ by $4$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 \cdot 64 y' x^{3} + 192 y' (x^{2} y) + 4 y' (12 x y^{2}) + 4 y' y^{3} - 3 y' = 0$

Step 5.2.2.9.3

Multiply $12$ by $4$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 4 \cdot 64 y' x^{3} + 192 y' (x^{2} y) + 48 y' (x y^{2}) + 4 y' y^{3} - 3 y' = 0$

Step 5.2.2.10

Multiply $4$ by $64$ .

$1024 x^{3} + 768 x^{2} y + 192 x y^{2} + 16 y^{3} + 256 y' x^{3} + 192 y' x^{2} y + 48 y' x y^{2} + 4 y' y^{3} - 3 y' = 0$

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 $1024 x^{3}$ from both sides of the equation.

$768 x^{2} y + 192 x y^{2} + 16 y^{3} + 256 y' x^{3} + 192 y' x^{2} y + 48 y' x y^{2} + 4 y' y^{3} - 3 y' = - 1024 x^{3}$

Step 5.3.2

Subtract $768 x^{2} y$ from both sides of the equation.

$192 x y^{2} + 16 y^{3} + 256 y' x^{3} + 192 y' x^{2} y + 48 y' x y^{2} + 4 y' y^{3} - 3 y' = - 1024 x^{3} - 768 x^{2} y$

Step 5.3.3

Subtract $192 x y^{2}$ from both sides of the equation.

$16 y^{3} + 256 y' x^{3} + 192 y' x^{2} y + 48 y' x y^{2} + 4 y' y^{3} - 3 y' = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2}$

Step 5.3.4

Subtract $16 y^{3}$ from both sides of the equation.

$256 y' x^{3} + 192 y' x^{2} y + 48 y' x y^{2} + 4 y' y^{3} - 3 y' = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4

Factor $y'$ out of $256 y' x^{3} + 192 y' x^{2} y + 48 y' x y^{2} + 4 y' y^{3} - 3 y'$ .

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Step 5.4.1

Factor $y'$ out of $256 y' x^{3}$ .

$y' (256 x^{3}) + 192 y' x^{2} y + 48 y' x y^{2} + 4 y' y^{3} - 3 y' = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.2

Factor $y'$ out of $192 y' x^{2} y$ .

$y' (256 x^{3}) + y' (192 x^{2} y) + 48 y' x y^{2} + 4 y' y^{3} - 3 y' = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.3

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

$y' (256 x^{3}) + y' (192 x^{2} y) + y' (48 x y^{2}) + 4 y' y^{3} - 3 y' = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.4

Factor $y'$ out of $4 y' y^{3}$ .

$y' (256 x^{3}) + y' (192 x^{2} y) + y' (48 x y^{2}) + y' (4 y^{3}) - 3 y' = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.5

Factor $y'$ out of $- 3 y'$ .

$y' (256 x^{3}) + y' (192 x^{2} y) + y' (48 x y^{2}) + y' (4 y^{3}) + y' \cdot - 3 = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.6

Factor $y'$ out of $y' (256 x^{3}) + y' (192 x^{2} y)$ .

$y' (256 x^{3} + 192 x^{2} y) + y' (48 x y^{2}) + y' (4 y^{3}) + y' \cdot - 3 = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.7

Factor $y'$ out of $y' (256 x^{3} + 192 x^{2} y) + y' (48 x y^{2})$ .

$y' (256 x^{3} + 192 x^{2} y + 48 x y^{2}) + y' (4 y^{3}) + y' \cdot - 3 = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.8

Factor $y'$ out of $y' (256 x^{3} + 192 x^{2} y + 48 x y^{2}) + y' (4 y^{3})$ .

$y' (256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3}) + y' \cdot - 3 = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.4.9

Factor $y'$ out of $y' (256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3}) + y' \cdot - 3$ .

$y' (256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3) = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$

Step 5.5

Divide each term in $y' (256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3) = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$ by $256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3$ and simplify.

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Step 5.5.1

Divide each term in $y' (256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3) = - 1024 x^{3} - 768 x^{2} y - 192 x y^{2} - 16 y^{3}$ by $256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3$ .

$\frac{y' (256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3)}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} = \frac{- 1024 x^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 768 x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 1024 x^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 768 x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Simplify terms.

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Step 5.5.3.1.1

Simplify each term.

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Step 5.5.3.1.1.1

Move the negative in front of the fraction.

$y' = - \frac{(1024) x^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 768 x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.1.2

Move the negative in front of the fraction.

$y' = - \frac{1024 x^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{(768) x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.1.3

Move the negative in front of the fraction.

$y' = - \frac{1024 x^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{768 x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{(192) x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} + \frac{- 16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.1.4

Move the negative in front of the fraction.

$y' = - \frac{1024 x^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{768 x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.2

Simplify terms.

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Step 5.5.3.1.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 1024 x^{3} - 768 x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.2.2

Factor $256 x^{2}$ out of $- 1024 x^{3} - 768 x^{2} y$ .

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Step 5.5.3.1.2.2.1

Factor $256 x^{2}$ out of $- 1024 x^{3}$ .

$y' = \frac{256 x^{2} (- 4 x) - 768 x^{2} y}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.2.2.2

Factor $256 x^{2}$ out of $- 768 x^{2} y$ .

$y' = \frac{256 x^{2} (- 4 x) + 256 x^{2} (- 3 y)}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.2.2.3

Factor $256 x^{2}$ out of $256 x^{2} (- 4 x) + 256 x^{2} (- 3 y)$ .

$y' = \frac{256 x^{2} (- 4 x - 3 y)}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.1.2.3

Combine the numerators over the common denominator.

$y' = \frac{256 x^{2} (- 4 x - 3 y) - 192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2

Simplify the numerator.

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Step 5.5.3.2.1

Factor $64 x$ out of $256 x^{2} (- 4 x - 3 y) - 192 x y^{2}$ .

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Step 5.5.3.2.1.1

Factor $64 x$ out of $256 x^{2} (- 4 x - 3 y)$ .

$y' = \frac{64 x (4 x (- 4 x - 3 y)) - 192 x y^{2}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.1.2

Factor $64 x$ out of $- 192 x y^{2}$ .

$y' = \frac{64 x (4 x (- 4 x - 3 y)) + 64 x (- 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.1.3

Factor $64 x$ out of $64 x (4 x (- 4 x - 3 y)) + 64 x (- 3 y^{2})$ .

$y' = \frac{64 x (4 x (- 4 x - 3 y) - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.2

Apply the distributive property.

$y' = \frac{64 x (4 x (- 4 x) + 4 x (- 3 y) - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.3

Rewrite using the commutative property of multiplication.

$y' = \frac{64 x (4 \cdot - 4 x \cdot x + 4 x (- 3 y) - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.4

Rewrite using the commutative property of multiplication.

$y' = \frac{64 x (4 \cdot - 4 x \cdot x + 4 \cdot - 3 x y - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.5

Simplify each term.

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Step 5.5.3.2.5.1

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

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Step 5.5.3.2.5.1.1

Move $x$ .

$y' = \frac{64 x (4 \cdot - 4 (x \cdot x) + 4 \cdot - 3 x y - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.5.1.2

Multiply $x$ by $x$ .

$y' = \frac{64 x (4 \cdot - 4 x^{2} + 4 \cdot - 3 x y - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.5.2

Multiply $4$ by $- 4$ .

$y' = \frac{64 x (- 16 x^{2} + 4 \cdot - 3 x y - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.2.5.3

Multiply $4$ by $- 3$ .

$y' = \frac{64 x (- 16 x^{2} - 12 x y - 3 y^{2})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3} - \frac{16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.3

Combine the numerators over the common denominator.

$y' = \frac{64 x (- 16 x^{2} - 12 x y - 3 y^{2}) - 16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4

Simplify the numerator.

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Step 5.5.3.4.1

Factor $16$ out of $64 x (- 16 x^{2} - 12 x y - 3 y^{2}) - 16 y^{3}$ .

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Step 5.5.3.4.1.1

Factor $16$ out of $64 x (- 16 x^{2} - 12 x y - 3 y^{2})$ .

$y' = \frac{16 (4 x (- 16 x^{2} - 12 x y - 3 y^{2})) - 16 y^{3}}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.1.2

Factor $16$ out of $- 16 y^{3}$ .

$y' = \frac{16 (4 x (- 16 x^{2} - 12 x y - 3 y^{2})) + 16 (- y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.1.3

Factor $16$ out of $16 (4 x (- 16 x^{2} - 12 x y - 3 y^{2})) + 16 (- y^{3})$ .

$y' = \frac{16 (4 x (- 16 x^{2} - 12 x y - 3 y^{2}) - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.2

Apply the distributive property.

$y' = \frac{16 (4 x (- 16 x^{2}) + 4 x (- 12 x y) + 4 x (- 3 y^{2}) - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.3

Simplify.

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Step 5.5.3.4.3.1

Rewrite using the commutative property of multiplication.

$y' = \frac{16 (4 \cdot - 16 x \cdot x^{2} + 4 x (- 12 x y) + 4 x (- 3 y^{2}) - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.3.2

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

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Step 5.5.3.4.3.2.1

Move $x$ .

$y' = \frac{16 (4 \cdot - 16 x \cdot x^{2} + 4 (x \cdot x) (- 12 y) + 4 x (- 3 y^{2}) - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.3.2.2

Multiply $x$ by $x$ .

$y' = \frac{16 (4 \cdot - 16 x \cdot x^{2} + 4 x^{2} (- 12 y) + 4 x (- 3 y^{2}) - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.3.3

Rewrite using the commutative property of multiplication.

$y' = \frac{16 (4 \cdot - 16 x \cdot x^{2} + 4 x^{2} (- 12 y) + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4

Simplify each term.

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Step 5.5.3.4.4.1

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

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Step 5.5.3.4.4.1.1

Move $x^{2}$ .

$y' = \frac{16 (4 \cdot - 16 (x^{2} x) + 4 x^{2} (- 12 y) + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4.1.2

Multiply $x^{2}$ by $x$ .

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Step 5.5.3.4.4.1.2.1

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

$y' = \frac{16 (4 \cdot - 16 (x^{2} x^{1}) + 4 x^{2} (- 12 y) + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4.1.2.2

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

$y' = \frac{16 (4 \cdot - 16 x^{2 + 1} + 4 x^{2} (- 12 y) + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4.1.3

Add $2$ and $1$ .

$y' = \frac{16 (4 \cdot - 16 x^{3} + 4 x^{2} (- 12 y) + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4.2

Multiply $4$ by $- 16$ .

$y' = \frac{16 (- 64 x^{3} + 4 x^{2} (- 12 y) + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4.3

Rewrite using the commutative property of multiplication.

$y' = \frac{16 (- 64 x^{3} + 4 \cdot - 12 x^{2} y + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4.4

Multiply $4$ by $- 12$ .

$y' = \frac{16 (- 64 x^{3} - 48 x^{2} y + 4 \cdot - 3 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.4.4.5

Multiply $4$ by $- 3$ .

$y' = \frac{16 (- 64 x^{3} - 48 x^{2} y - 12 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5

Simplify with factoring out.

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Step 5.5.3.5.1

Factor $- 1$ out of $- 64 x^{3}$ .

$y' = \frac{16 (- (64 x^{3}) - 48 x^{2} y - 12 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.2

Factor $- 1$ out of $- 48 x^{2} y$ .

$y' = \frac{16 (- (64 x^{3}) - (48 x^{2} y) - 12 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.3

Factor $- 1$ out of $- (64 x^{3}) - (48 x^{2} y)$ .

$y' = \frac{16 (- (64 x^{3} + 48 x^{2} y) - 12 x y^{2} - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.4

Factor $- 1$ out of $- 12 x y^{2}$ .

$y' = \frac{16 (- (64 x^{3} + 48 x^{2} y) - (12 x y^{2}) - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.5

Factor $- 1$ out of $- (64 x^{3} + 48 x^{2} y) - (12 x y^{2})$ .

$y' = \frac{16 (- (64 x^{3} + 48 x^{2} y + 12 x y^{2}) - y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.6

Factor $- 1$ out of $- y^{3}$ .

$y' = \frac{16 (- (64 x^{3} + 48 x^{2} y + 12 x y^{2}) - (y^{3}))}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.7

Factor $- 1$ out of $- (64 x^{3} + 48 x^{2} y + 12 x y^{2}) - (y^{3})$ .

$y' = \frac{16 (- (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3}))}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.8

Simplify the expression.

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Step 5.5.3.5.8.1

Rewrite $- (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3})$ as $- 1 (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3})$ .

$y' = \frac{16 (- 1 (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3}))}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 5.5.3.5.8.2

Move the negative in front of the fraction.

$y' = - \frac{16 (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$

Step 6

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

$\frac{d y}{d x} = - \frac{16 (64 x^{3} + 48 x^{2} y + 12 x y^{2} + y^{3})}{256 x^{3} + 192 x^{2} y + 48 x y^{2} + 4 y^{3} - 3}$