Calculus Examples

$6 {(x^{2} + y^{2})}^{2} = 169 (x^{2} - y^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (6 {(x^{2} + y^{2})}^{2}) = \frac{d}{d x} (169 (x^{2} - y^{2}))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 {(x^{2} + y^{2})}^{2}$ with respect to $x$ is $6 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2} + y^{2}$ .

$6 (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.2.2

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

$6 (2 u_{1} \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.2.3

Replace all occurrences of $u_{1}$ with $x^{2} + y^{2}$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $2$ by $6$ .

$12 ((x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.3.2

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

$12 (x^{2} + y^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

Step 2.3.3

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

$12 (x^{2} + y^{2}) (2 x + \frac{d}{d x} [y^{2}])$

Step 2.4

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

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Step 2.4.1

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

$12 (x^{2} + y^{2}) (2 x + \frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [y])$

Step 2.4.2

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

$12 (x^{2} + y^{2}) (2 x + 2 u_{2} \frac{d}{d x} [y])$

Step 2.4.3

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

$12 (x^{2} + y^{2}) (2 x + 2 y \frac{d}{d x} [y])$

Step 2.5

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

$12 (x^{2} + y^{2}) (2 x + 2 y y')$

Step 2.6

Apply the distributive property.

$(12 x^{2} + 12 y^{2}) (2 x + 2 y y')$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

Since $169$ is constant with respect to $x$ , the derivative of $169 (x^{2} - y^{2})$ with respect to $x$ is $169 \frac{d}{d x} [x^{2} - y^{2}]$ .

$169 \frac{d}{d x} [x^{2} - y^{2}]$

Step 3.1.2

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

$169 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- y^{2}])$

Step 3.1.3

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

$169 (2 x + \frac{d}{d x} [- y^{2}])$

Step 3.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{2}$ with respect to $x$ is $- \frac{d}{d x} [y^{2}]$ .

$169 (2 x - \frac{d}{d x} [y^{2}])$

Step 3.2

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

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Step 3.2.1

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

$169 (2 x - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]))$

Step 3.2.2

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

$169 (2 x - (2 u \frac{d}{d x} [y]))$

Step 3.2.3

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

$169 (2 x - (2 y \frac{d}{d x} [y]))$

Step 3.3

Multiply $2$ by $- 1$ .

$169 (2 x - 2 (y \frac{d}{d x} [y]))$

Step 3.4

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

$169 (2 x - 2 y y')$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$169 (2 x) + 169 (- 2 y y')$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Multiply $2$ by $169$ .

$338 x + 169 (- 2 y y')$

Step 3.5.2.2

Multiply $- 2$ by $169$ .

$338 x - 338 y y'$

Step 3.5.3

Reorder terms.

$- 338 y y' + 338 x$

Step 4

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

$(12 x^{2} + 12 y^{2}) (2 x + 2 y y') = - 338 y y' + 338 x$

Step 5

Solve for $y'$ .

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Step 5.1

Simplify $(12 x^{2} + 12 y^{2}) (2 x + 2 y y')$ .

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Step 5.1.1

Rewrite.

$0 + 0 + (12 x^{2} + 12 y^{2}) (2 x + 2 y y') = - 338 y y' + 338 x$

Step 5.1.2

Simplify by adding zeros.

$(12 x^{2} + 12 y^{2}) (2 x + 2 y y') = - 338 y y' + 338 x$

Step 5.1.3

Expand $(12 x^{2} + 12 y^{2}) (2 x + 2 y y')$ using the FOIL Method.

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Step 5.1.3.1

Apply the distributive property.

$12 x^{2} (2 x + 2 y y') + 12 y^{2} (2 x + 2 y y') = - 338 y y' + 338 x$

Step 5.1.3.2

Apply the distributive property.

$12 x^{2} (2 x) + 12 x^{2} (2 y y') + 12 y^{2} (2 x + 2 y y') = - 338 y y' + 338 x$

Step 5.1.3.3

Apply the distributive property.

$12 x^{2} (2 x) + 12 x^{2} (2 y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4

Simplify each term.

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Step 5.1.4.1

Rewrite using the commutative property of multiplication.

$12 \cdot 2 x^{2} x + 12 x^{2} (2 y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.2

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

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Step 5.1.4.2.1

Move $x$ .

$12 \cdot 2 (x \cdot x^{2}) + 12 x^{2} (2 y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.2.2

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

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Step 5.1.4.2.2.1

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

$12 \cdot 2 (x^{1} x^{2}) + 12 x^{2} (2 y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.2.2.2

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

$12 \cdot 2 x^{1 + 2} + 12 x^{2} (2 y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.2.3

Add $1$ and $2$ .

$12 \cdot 2 x^{3} + 12 x^{2} (2 y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.3

Multiply $12$ by $2$ .

$24 x^{3} + 12 x^{2} (2 y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.4

Multiply $2$ by $12$ .

$24 x^{3} + 24 x^{2} (y y') + 12 y^{2} (2 x) + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.5

Rewrite using the commutative property of multiplication.

$24 x^{3} + 24 x^{2} y y' + 12 \cdot 2 y^{2} x + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.6

Multiply $12$ by $2$ .

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 12 y^{2} (2 y y') = - 338 y y' + 338 x$

Step 5.1.4.7

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

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Step 5.1.4.7.1

Move $y$ .

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 12 (y \cdot y^{2}) (2 y') = - 338 y y' + 338 x$

Step 5.1.4.7.2

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

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Step 5.1.4.7.2.1

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

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 12 (y^{1} y^{2}) (2 y') = - 338 y y' + 338 x$

Step 5.1.4.7.2.2

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

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 12 y^{1 + 2} (2 y') = - 338 y y' + 338 x$

Step 5.1.4.7.3

Add $1$ and $2$ .

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 12 y^{3} (2 y') = - 338 y y' + 338 x$

Step 5.1.4.8

Rewrite using the commutative property of multiplication.

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 12 \cdot 2 y^{3} y' = - 338 y y' + 338 x$

Step 5.1.4.9

Multiply $12$ by $2$ .

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 24 y^{3} y' = - 338 y y' + 338 x$

Step 5.2

Add $338 y y'$ to both sides of the equation.

$24 x^{3} + 24 x^{2} y y' + 24 y^{2} x + 24 y^{3} y' + 338 y y' = 338 x$

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 $24 x^{3}$ from both sides of the equation.

$24 x^{2} y y' + 24 y^{2} x + 24 y^{3} y' + 338 y y' = 338 x - 24 x^{3}$

Step 5.3.2

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

$24 x^{2} y y' + 24 y^{3} y' + 338 y y' = 338 x - 24 x^{3} - 24 y^{2} x$

Step 5.4

Factor $2 y y'$ out of $24 x^{2} y y' + 24 y^{3} y' + 338 y y'$ .

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Step 5.4.1

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

$2 y y' (12 x^{2}) + 24 y^{3} y' + 338 y y' = 338 x - 24 x^{3} - 24 y^{2} x$

Step 5.4.2

Factor $2 y y'$ out of $24 y^{3} y'$ .

$2 y y' (12 x^{2}) + 2 y y' (12 y^{2}) + 338 y y' = 338 x - 24 x^{3} - 24 y^{2} x$

Step 5.4.3

Factor $2 y y'$ out of $338 y y'$ .

$2 y y' (12 x^{2}) + 2 y y' (12 y^{2}) + 2 y y' \cdot 169 = 338 x - 24 x^{3} - 24 y^{2} x$

Step 5.4.4

Factor $2 y y'$ out of $2 y y' (12 x^{2}) + 2 y y' (12 y^{2})$ .

$2 y y' (12 x^{2} + 12 y^{2}) + 2 y y' \cdot 169 = 338 x - 24 x^{3} - 24 y^{2} x$

Step 5.4.5

Factor $2 y y'$ out of $2 y y' (12 x^{2} + 12 y^{2}) + 2 y y' \cdot 169$ .

$2 y y' (12 x^{2} + 12 y^{2} + 169) = 338 x - 24 x^{3} - 24 y^{2} x$

Step 5.5

Divide each term in $2 y y' (12 x^{2} + 12 y^{2} + 169) = 338 x - 24 x^{3} - 24 y^{2} x$ by $2 y (12 x^{2} + 12 y^{2} + 169)$ and simplify.

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Step 5.5.1

Divide each term in $2 y y' (12 x^{2} + 12 y^{2} + 169) = 338 x - 24 x^{3} - 24 y^{2} x$ by $2 y (12 x^{2} + 12 y^{2} + 169)$ .

$\frac{2 y y' (12 x^{2} + 12 y^{2} + 169)}{2 y (12 x^{2} + 12 y^{2} + 169)} = \frac{338 x}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $2$ .

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Step 5.5.2.1.1

Cancel the common factor.

Step 5.5.2.1.2

Rewrite the expression.

$\frac{y y' (12 x^{2} + 12 y^{2} + 169)}{y (12 x^{2} + 12 y^{2} + 169)} = \frac{338 x}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.2.2

Cancel the common factor of $y$ .

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Step 5.5.2.2.1

Cancel the common factor.

Step 5.5.2.2.2

Rewrite the expression.

$\frac{y' (12 x^{2} + 12 y^{2} + 169)}{12 x^{2} + 12 y^{2} + 169} = \frac{338 x}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.2.3

Cancel the common factor of $12 x^{2} + 12 y^{2} + 169$ .

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Step 5.5.2.3.1

Cancel the common factor.

Step 5.5.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{338 x}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

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 $338$ and $2$ .

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Step 5.5.3.1.1.1

Factor $2$ out of $338 x$ .

$y' = \frac{2 (169 x)}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.1.2

Cancel the common factors.

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Step 5.5.3.1.1.2.1

Factor $2$ out of $2 y (12 x^{2} + 12 y^{2} + 169)$ .

$y' = \frac{2 (169 x)}{2 (y (12 x^{2} + 12 y^{2} + 169))} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (169 x)}{2 (y (12 x^{2} + 12 y^{2} + 169))} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 x^{3}}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.2

Cancel the common factor of $- 24$ and $2$ .

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Step 5.5.3.1.2.1

Factor $2$ out of $- 24 x^{3}$ .

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{2 (- 12 x^{3})}{2 y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.2.2

Cancel the common factors.

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Step 5.5.3.1.2.2.1

Factor $2$ out of $2 y (12 x^{2} + 12 y^{2} + 169)$ .

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{2 (- 12 x^{3})}{2 (y (12 x^{2} + 12 y^{2} + 169))} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.2.2.2

Cancel the common factor.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{2 (- 12 x^{3})}{2 (y (12 x^{2} + 12 y^{2} + 169))} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.2.2.3

Rewrite the expression.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 24 y^{2} x}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.4

Cancel the common factor of $- 24$ and $2$ .

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Step 5.5.3.1.4.1

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

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{2 (- 12 y^{2} x)}{2 y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.4.2

Cancel the common factors.

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Step 5.5.3.1.4.2.1

Factor $2$ out of $2 y (12 x^{2} + 12 y^{2} + 169)$ .

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{2 (- 12 y^{2} x)}{2 (y (12 x^{2} + 12 y^{2} + 169))}$

Step 5.5.3.1.4.2.2

Cancel the common factor.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{2 (- 12 y^{2} x)}{2 (y (12 x^{2} + 12 y^{2} + 169))}$

Step 5.5.3.1.4.2.3

Rewrite the expression.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 12 y^{2} x}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.5

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

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Step 5.5.3.1.5.1

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

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{y (- 12 y x)}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.5.2

Cancel the common factors.

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Step 5.5.3.1.5.2.1

Cancel the common factor.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{y (- 12 y x)}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.1.5.2.2

Rewrite the expression.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{- 12 y x}{12 x^{2} + 12 y^{2} + 169}$

Step 5.5.3.1.6

Move the negative in front of the fraction.

$y' = \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 y x}{12 x^{2} + 12 y^{2} + 169}$

Step 5.5.3.2

To write $- \frac{12 y x}{12 x^{2} + 12 y^{2} + 169}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 y x}{12 x^{2} + 12 y^{2} + 169} \cdot \frac{y}{y}$

Step 5.5.3.3

Write each expression with a common denominator of $y (12 x^{2} + 12 y^{2} + 169)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.3.3.1

Multiply $\frac{12 y x}{12 x^{2} + 12 y^{2} + 169}$ by $\frac{y}{y}$ .

$y' = - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 y x y}{(12 x^{2} + 12 y^{2} + 169) y}$

Step 5.5.3.3.2

Reorder the factors of $(12 x^{2} + 12 y^{2} + 169) y$ .

$y' = - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{169 x}{y (12 x^{2} + 12 y^{2} + 169)} - \frac{12 y x y}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.4

Combine the numerators over the common denominator.

$y' = - \frac{12 x^{3}}{y (12 x^{2} + 12 y^{2} + 169)} + \frac{169 x - 12 y x y}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.5

Combine the numerators over the common denominator.

$y' = \frac{- 12 x^{3} + 169 x - 12 y x y}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.6

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

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Step 5.5.3.6.1

Move $y$ .

$y' = \frac{- 12 x^{3} + 169 x - 12 (y \cdot y) x}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.6.2

Multiply $y$ by $y$ .

$y' = \frac{- 12 x^{3} + 169 x - 12 y^{2} x}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.7

Factor $x$ out of $- 12 x^{3} + 169 x - 12 y^{2} x$ .

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Step 5.5.3.7.1

Factor $x$ out of $- 12 x^{3}$ .

$y' = \frac{x (- 12 x^{2}) + 169 x - 12 y^{2} x}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.7.2

Factor $x$ out of $169 x$ .

$y' = \frac{x (- 12 x^{2}) + x \cdot 169 - 12 y^{2} x}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.7.3

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

$y' = \frac{x (- 12 x^{2}) + x \cdot 169 + x (- 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.7.4

Factor $x$ out of $x (- 12 x^{2}) + x \cdot 169$ .

$y' = \frac{x (- 12 x^{2} + 169) + x (- 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.7.5

Factor $x$ out of $x (- 12 x^{2} + 169) + x (- 12 y^{2})$ .

$y' = \frac{x (- 12 x^{2} + 169 - 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.8

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

$y' = \frac{x (- (12 x^{2}) + 169 - 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.9

Rewrite $169$ as $- 1 (- 169)$ .

$y' = \frac{x (- (12 x^{2}) - 1 (- 169) - 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.10

Factor $- 1$ out of $- (12 x^{2}) - 1 (- 169)$ .

$y' = \frac{x (- (12 x^{2} - 169) - 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.11

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

$y' = \frac{x (- (12 x^{2} - 169) - (12 y^{2}))}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.12

Factor $- 1$ out of $- (12 x^{2} - 169) - (12 y^{2})$ .

$y' = \frac{x (- (12 x^{2} - 169 + 12 y^{2}))}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.13

Simplify the expression.

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Step 5.5.3.13.1

Rewrite $- (12 x^{2} - 169 + 12 y^{2})$ as $- 1 (12 x^{2} - 169 + 12 y^{2})$ .

$y' = \frac{x (- 1 (12 x^{2} - 169 + 12 y^{2}))}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 5.5.3.13.2

Move the negative in front of the fraction.

$y' = - \frac{x (12 x^{2} - 169 + 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$

Step 6

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

$\frac{d y}{d x} = - \frac{x (12 x^{2} - 169 + 12 y^{2})}{y (12 x^{2} + 12 y^{2} + 169)}$