Calculus Examples

$x^{3} = \frac{2 x - y}{x + 3 y}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$3 x^{2}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $2$ by $1$ .

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

Step 3.2.5

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

$\frac{(x + 3 y) (2 - y') - (2 x - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$\frac{(x + 3 y) (2 - y') + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2

Simplify the numerator.

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Step 3.8.2.1

Simplify each term.

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Step 3.8.2.1.1

Expand $(x + 3 y) (2 - y')$ using the FOIL Method.

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Step 3.8.2.1.1.1

Apply the distributive property.

$\frac{x (2 - y') + 3 y (2 - y') + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.1.2

Apply the distributive property.

$\frac{x \cdot 2 + x (- y') + 3 y (2 - y') + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.1.3

Apply the distributive property.

$\frac{x \cdot 2 + x (- y') + 3 y \cdot 2 + 3 y (- y') + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.2

Simplify each term.

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Step 3.8.2.1.2.1

Move $2$ to the left of $x$ .

$\frac{2 \cdot x + x (- y') + 3 y \cdot 2 + 3 y (- y') + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.2.2

Rewrite using the commutative property of multiplication.

$\frac{2 x - x y' + 3 y \cdot 2 + 3 y (- y') + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.2.3

Multiply $2$ by $3$ .

$\frac{2 x - x y' + 6 y + 3 y (- y') + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.2.4

Multiply $- 1$ by $3$ .

$\frac{2 x - x y' + 6 y - 3 y y' + (- (2 x) - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.3

Simplify each term.

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Step 3.8.2.1.3.1

Multiply $2$ by $- 1$ .

$\frac{2 x - x y' + 6 y - 3 y y' + (- 2 x - - y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.3.2

Multiply $- - y$ .

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Step 3.8.2.1.3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x - x y' + 6 y - 3 y y' + (- 2 x + 1 y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.3.2.2

Multiply $y$ by $1$ .

$\frac{2 x - x y' + 6 y - 3 y y' + (- 2 x + y) (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.4

Expand $(- 2 x + y) (1 + 3 y')$ using the FOIL Method.

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Step 3.8.2.1.4.1

Apply the distributive property.

$\frac{2 x - x y' + 6 y - 3 y y' - 2 x (1 + 3 y') + y (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.4.2

Apply the distributive property.

$\frac{2 x - x y' + 6 y - 3 y y' - 2 x \cdot 1 - 2 x (3 y') + y (1 + 3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.4.3

Apply the distributive property.

$\frac{2 x - x y' + 6 y - 3 y y' - 2 x \cdot 1 - 2 x (3 y') + y \cdot 1 + y (3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.5

Simplify each term.

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Step 3.8.2.1.5.1

Multiply $- 2$ by $1$ .

$\frac{2 x - x y' + 6 y - 3 y y' - 2 x - 2 x (3 y') + y \cdot 1 + y (3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.5.2

Multiply $3$ by $- 2$ .

$\frac{2 x - x y' + 6 y - 3 y y' - 2 x - 6 x y' + y \cdot 1 + y (3 y')}{{(x + 3 y)}^{2}}$

Step 3.8.2.1.5.3

Multiply $y$ by $1$ .

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

Step 3.8.2.1.5.4

Rewrite using the commutative property of multiplication.

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

Step 3.8.2.2

Combine the opposite terms in $2 x - x y' + 6 y - 3 y y' - 2 x - 6 x y' + y + 3 y y'$ .

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Step 3.8.2.2.1

Subtract $2 x$ from $2 x$ .

$\frac{- x y' + 6 y - 3 y y' + 0 - 6 x y' + y + 3 y y'}{{(x + 3 y)}^{2}}$

Step 3.8.2.2.2

Add $- x y' + 6 y - 3 y y'$ and $0$ .

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

Step 3.8.2.2.3

Add $- 3 y y'$ and $3 y y'$ .

$\frac{- x y' + 6 y - 6 x y' + y + 0}{{(x + 3 y)}^{2}}$

Step 3.8.2.2.4

Add $- x y' + 6 y - 6 x y' + y$ and $0$ .

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

Step 3.8.2.3

Subtract $6 x y'$ from $- x y'$ .

$\frac{- 7 x y' + 6 y + y}{{(x + 3 y)}^{2}}$

Step 3.8.2.4

Add $6 y$ and $y$ .

$\frac{- 7 x y' + 7 y}{{(x + 3 y)}^{2}}$

Step 3.8.3

Factor $7$ out of $- 7 x y' + 7 y$ .

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Step 3.8.3.1

Factor $7$ out of $- 7 x y'$ .

$\frac{7 (- x y') + 7 y}{{(x + 3 y)}^{2}}$

Step 3.8.3.2

Factor $7$ out of $7 y$ .

$\frac{7 (- x y') + 7 (y)}{{(x + 3 y)}^{2}}$

Step 3.8.3.3

Factor $7$ out of $7 (- x y') + 7 (y)$ .

$\frac{7 (- x y' + y)}{{(x + 3 y)}^{2}}$

Step 3.8.4

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

$\frac{7 (- (x y') + y)}{{(x + 3 y)}^{2}}$

Step 3.8.5

Factor $- 1$ out of $y$ .

$\frac{7 (- (x y') - 1 (- y))}{{(x + 3 y)}^{2}}$

Step 3.8.6

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

$\frac{7 (- (x y' - y))}{{(x + 3 y)}^{2}}$

Step 3.8.7

Rewrite $- (x y' - y)$ as $- 1 (x y' - y)$ .

$\frac{7 (- 1 (x y' - y))}{{(x + 3 y)}^{2}}$

Step 3.8.8

Move the negative in front of the fraction.

$- \frac{7 (x y' - y)}{{(x + 3 y)}^{2}}$

Step 4

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

$3 x^{2} = - \frac{7 (x y' - y)}{{(x + 3 y)}^{2}}$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- \frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = 3 x^{2}$ .

$- \frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = 3 x^{2}$

Step 5.2

Divide each term in $- \frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = 3 x^{2}$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = 3 x^{2}$ by $- 1$ .

$\frac{- \frac{7 (x y' - y)}{{(x + 3 y)}^{2}}}{- 1} = \frac{3 x^{2}}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{7 (x y' - y)}{{(x + 3 y)}^{2}}}{1} = \frac{3 x^{2}}{- 1}$

Step 5.2.2.2

Divide $\frac{7 (x y' - y)}{{(x + 3 y)}^{2}}$ by $1$ .

$\frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = \frac{3 x^{2}}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative one from the denominator of $\frac{3 x^{2}}{- 1}$ .

$\frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = - 1 \cdot (3 x^{2})$

Step 5.2.3.2

Rewrite $- 1 \cdot (3 x^{2})$ as $- (3 x^{2})$ .

$\frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = - (3 x^{2})$

Step 5.2.3.3

Multiply $3$ by $- 1$ .

$\frac{7 (x y' - y)}{{(x + 3 y)}^{2}} = - 3 x^{2}$

Step 5.3

Multiply both sides by ${(x + 3 y)}^{2}$ .

$\frac{7 (x y' - y)}{{(x + 3 y)}^{2}} {(x + 3 y)}^{2} = - 3 x^{2} {(x + 3 y)}^{2}$

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $\frac{7 (x y' - y)}{{(x + 3 y)}^{2}} {(x + 3 y)}^{2}$ .

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Step 5.4.1.1

Cancel the common factor of ${(x + 3 y)}^{2}$ .

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Step 5.4.1.1.1

Cancel the common factor.

$\frac{7 (x y' - y)}{{(x + 3 y)}^{2}} {(x + 3 y)}^{2} = - 3 x^{2} {(x + 3 y)}^{2}$

Step 5.4.1.1.2

Rewrite the expression.

$7 (x y' - y) = - 3 x^{2} {(x + 3 y)}^{2}$

Step 5.4.1.2

Apply the distributive property.

$7 (x y') + 7 (- y) = - 3 x^{2} {(x + 3 y)}^{2}$

Step 5.4.1.3

Simplify the expression.

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Step 5.4.1.3.1

Multiply $- 1$ by $7$ .

$7 x y' - 7 y = - 3 x^{2} {(x + 3 y)}^{2}$

Step 5.4.1.3.2

Move $x$ .

$7 y' x - 7 y = - 3 x^{2} {(x + 3 y)}^{2}$

Step 5.5

Solve for $y'$ .

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Step 5.5.1

Simplify $- 3 x^{2} {(x + 3 y)}^{2}$ .

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Step 5.5.1.1

Rewrite.

$7 y' x - 7 y = 0 + 0 - 3 x^{2} {(x + 3 y)}^{2}$

Step 5.5.1.2

Rewrite ${(x + 3 y)}^{2}$ as $(x + 3 y) (x + 3 y)$ .

$7 y' x - 7 y = - 3 x^{2} ((x + 3 y) (x + 3 y))$

Step 5.5.1.3

Expand $(x + 3 y) (x + 3 y)$ using the FOIL Method.

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Step 5.5.1.3.1

Apply the distributive property.

$7 y' x - 7 y = - 3 x^{2} (x (x + 3 y) + 3 y (x + 3 y))$

Step 5.5.1.3.2

Apply the distributive property.

$7 y' x - 7 y = - 3 x^{2} (x \cdot x + x (3 y) + 3 y (x + 3 y))$

Step 5.5.1.3.3

Apply the distributive property.

$7 y' x - 7 y = - 3 x^{2} (x \cdot x + x (3 y) + 3 y x + 3 y (3 y))$

Step 5.5.1.4

Simplify and combine like terms.

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Step 5.5.1.4.1

Simplify each term.

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Step 5.5.1.4.1.1

Multiply $x$ by $x$ .

$7 y' x - 7 y = - 3 x^{2} (x^{2} + x (3 y) + 3 y x + 3 y (3 y))$

Step 5.5.1.4.1.2

Rewrite using the commutative property of multiplication.

$7 y' x - 7 y = - 3 x^{2} (x^{2} + 3 x y + 3 y x + 3 y (3 y))$

Step 5.5.1.4.1.3

Rewrite using the commutative property of multiplication.

$7 y' x - 7 y = - 3 x^{2} (x^{2} + 3 x y + 3 y x + 3 \cdot 3 y \cdot y)$

Step 5.5.1.4.1.4

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

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Step 5.5.1.4.1.4.1

Move $y$ .

$7 y' x - 7 y = - 3 x^{2} (x^{2} + 3 x y + 3 y x + 3 \cdot 3 (y \cdot y))$

Step 5.5.1.4.1.4.2

Multiply $y$ by $y$ .

$7 y' x - 7 y = - 3 x^{2} (x^{2} + 3 x y + 3 y x + 3 \cdot 3 y^{2})$

Step 5.5.1.4.1.5

Multiply $3$ by $3$ .

$7 y' x - 7 y = - 3 x^{2} (x^{2} + 3 x y + 3 y x + 9 y^{2})$

Step 5.5.1.4.2

Add $3 x y$ and $3 y x$ .

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Step 5.5.1.4.2.1

Move $y$ .

$7 y' x - 7 y = - 3 x^{2} (x^{2} + 3 x y + 3 x y + 9 y^{2})$

Step 5.5.1.4.2.2

Add $3 x y$ and $3 x y$ .

$7 y' x - 7 y = - 3 x^{2} (x^{2} + 6 x y + 9 y^{2})$

Step 5.5.1.5

Apply the distributive property.

$7 y' x - 7 y = - 3 x^{2} x^{2} - 3 x^{2} (6 x y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6

Simplify.

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Step 5.5.1.6.1

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

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Step 5.5.1.6.1.1

Move $x^{2}$ .

$7 y' x - 7 y = - 3 (x^{2} x^{2}) - 3 x^{2} (6 x y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6.1.2

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

$7 y' x - 7 y = - 3 x^{2 + 2} - 3 x^{2} (6 x y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6.1.3

Add $2$ and $2$ .

$7 y' x - 7 y = - 3 x^{4} - 3 x^{2} (6 x y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6.2

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

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Step 5.5.1.6.2.1

Move $x$ .

$7 y' x - 7 y = - 3 x^{4} - 3 (x \cdot x^{2}) (6 y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6.2.2

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

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Step 5.5.1.6.2.2.1

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

$7 y' x - 7 y = - 3 x^{4} - 3 (x^{1} x^{2}) (6 y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6.2.2.2

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

$7 y' x - 7 y = - 3 x^{4} - 3 x^{1 + 2} (6 y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6.2.3

Add $1$ and $2$ .

$7 y' x - 7 y = - 3 x^{4} - 3 x^{3} (6 y) - 3 x^{2} (9 y^{2})$

Step 5.5.1.6.3

Rewrite using the commutative property of multiplication.

$7 y' x - 7 y = - 3 x^{4} - 3 x^{3} (6 y) - 3 \cdot 9 x^{2} y^{2}$

Step 5.5.1.7

Simplify each term.

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Step 5.5.1.7.1

Rewrite using the commutative property of multiplication.

$7 y' x - 7 y = - 3 x^{4} - 3 \cdot 6 x^{3} y - 3 \cdot 9 x^{2} y^{2}$

Step 5.5.1.7.2

Multiply $- 3$ by $6$ .

$7 y' x - 7 y = - 3 x^{4} - 18 x^{3} y - 3 \cdot 9 x^{2} y^{2}$

Step 5.5.1.7.3

Multiply $- 3$ by $9$ .

$7 y' x - 7 y = - 3 x^{4} - 18 x^{3} y - 27 x^{2} y^{2}$

Step 5.5.2

Add $7 y$ to both sides of the equation.

$7 y' x = - 3 x^{4} - 18 x^{3} y - 27 x^{2} y^{2} + 7 y$

Step 5.5.3

Divide each term in $7 y' x = - 3 x^{4} - 18 x^{3} y - 27 x^{2} y^{2} + 7 y$ by $7 x$ and simplify.

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Step 5.5.3.1

Divide each term in $7 y' x = - 3 x^{4} - 18 x^{3} y - 27 x^{2} y^{2} + 7 y$ by $7 x$ .

$\frac{7 y' x}{7 x} = \frac{- 3 x^{4}}{7 x} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.2

Simplify the left side.

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Step 5.5.3.2.1

Cancel the common factor of $7$ .

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Step 5.5.3.2.1.1

Cancel the common factor.

$\frac{7 y' x}{7 x} = \frac{- 3 x^{4}}{7 x} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.2.1.2

Rewrite the expression.

$\frac{y' x}{x} = \frac{- 3 x^{4}}{7 x} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.2.2

Cancel the common factor of $x$ .

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Step 5.5.3.2.2.1

Cancel the common factor.

$\frac{y' x}{x} = \frac{- 3 x^{4}}{7 x} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 3 x^{4}}{7 x} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3

Simplify the right side.

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Step 5.5.3.3.1

Simplify each term.

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Step 5.5.3.3.1.1

Cancel the common factor of $x^{4}$ and $x$ .

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Step 5.5.3.3.1.1.1

Factor $x$ out of $- 3 x^{4}$ .

$y' = \frac{x (- 3 x^{3})}{7 x} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.1.2

Cancel the common factors.

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Step 5.5.3.3.1.1.2.1

Factor $x$ out of $7 x$ .

$y' = \frac{x (- 3 x^{3})}{x \cdot 7} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.1.2.2

Cancel the common factor.

$y' = \frac{x (- 3 x^{3})}{x \cdot 7} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.1.2.3

Rewrite the expression.

$y' = \frac{- 3 x^{3}}{7} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{3 x^{3}}{7} + \frac{- 18 x^{3} y}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.3

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.5.3.3.1.3.1

Factor $x$ out of $- 18 x^{3} y$ .

$y' = - \frac{3 x^{3}}{7} + \frac{x (- 18 x^{2} y)}{7 x} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.3.2

Cancel the common factors.

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Step 5.5.3.3.1.3.2.1

Factor $x$ out of $7 x$ .

$y' = - \frac{3 x^{3}}{7} + \frac{x (- 18 x^{2} y)}{x \cdot 7} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.3.2.2

Cancel the common factor.

$y' = - \frac{3 x^{3}}{7} + \frac{x (- 18 x^{2} y)}{x \cdot 7} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.3.2.3

Rewrite the expression.

$y' = - \frac{3 x^{3}}{7} + \frac{- 18 x^{2} y}{7} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} + \frac{- 27 x^{2} y^{2}}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.5

Cancel the common factor of $x^{2}$ and $x$ .

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Step 5.5.3.3.1.5.1

Factor $x$ out of $- 27 x^{2} y^{2}$ .

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} + \frac{x (- 27 x y^{2})}{7 x} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.5.2

Cancel the common factors.

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Step 5.5.3.3.1.5.2.1

Factor $x$ out of $7 x$ .

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} + \frac{x (- 27 x y^{2})}{x \cdot 7} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.5.2.2

Cancel the common factor.

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} + \frac{x (- 27 x y^{2})}{x \cdot 7} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.5.2.3

Rewrite the expression.

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} + \frac{- 27 x y^{2}}{7} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.6

Move the negative in front of the fraction.

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} - \frac{27 x y^{2}}{7} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.7

Cancel the common factor of $7$ .

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Step 5.5.3.3.1.7.1

Cancel the common factor.

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} - \frac{27 x y^{2}}{7} + \frac{7 y}{7 x}$

Step 5.5.3.3.1.7.2

Rewrite the expression.

$y' = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} - \frac{27 x y^{2}}{7} + \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = - \frac{3 x^{3}}{7} - \frac{18 x^{2} y}{7} - \frac{27 x y^{2}}{7} + \frac{y}{x}$