Calculus Examples

$16 x^{4} + 32 x^{2} y^{2} + 16 y^{4} = 144 x^{2} - 144 y^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (16 x^{4} + 32 x^{2} y^{2} + 16 y^{4}) = \frac{d}{d x} (144 x^{2} - 144 y^{2})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $16 x^{4} + 32 x^{2} y^{2} + 16 y^{4}$ with respect to $x$ is $\frac{d}{d x} [16 x^{4}] + \frac{d}{d x} [32 x^{2} y^{2}] + \frac{d}{d x} [16 y^{4}]$ .

$\frac{d}{d x} [16 x^{4}] + \frac{d}{d x} [32 x^{2} y^{2}] + \frac{d}{d x} [16 y^{4}]$

Step 2.2

Evaluate $\frac{d}{d x} [16 x^{4}]$ .

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Step 2.2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{4}$ with respect to $x$ is $16 \frac{d}{d x} [x^{4}]$ .

$16 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [32 x^{2} y^{2}] + \frac{d}{d x} [16 y^{4}]$

Step 2.2.2

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

$16 (4 x^{3}) + \frac{d}{d x} [32 x^{2} y^{2}] + \frac{d}{d x} [16 y^{4}]$

Step 2.2.3

Multiply $4$ by $16$ .

$64 x^{3} + \frac{d}{d x} [32 x^{2} y^{2}] + \frac{d}{d x} [16 y^{4}]$

Step 2.3

Evaluate $\frac{d}{d x} [32 x^{2} y^{2}]$ .

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Step 2.3.1

Since $32$ is constant with respect to $x$ , the derivative of $32 x^{2} y^{2}$ with respect to $x$ is $32 \frac{d}{d x} [x^{2} y^{2}]$ .

$64 x^{3} + 32 \frac{d}{d x} [x^{2} y^{2}] + \frac{d}{d x} [16 y^{4}]$

Step 2.3.2

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

$64 x^{3} + 32 (x^{2} \frac{d}{d x} [y^{2}] + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [16 y^{4}]$

Step 2.3.3

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 2.3.3.1

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

$64 x^{3} + 32 (x^{2} (\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [16 y^{4}]$

Step 2.3.3.2

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

$64 x^{3} + 32 (x^{2} (2 u_{1} \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [16 y^{4}]$

Step 2.3.3.3

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

$64 x^{3} + 32 (x^{2} (2 y \frac{d}{d x} [y]) + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [16 y^{4}]$

Step 2.3.4

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

$64 x^{3} + 32 (x^{2} (2 y y') + y^{2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [16 y^{4}]$

Step 2.3.5

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

$64 x^{3} + 32 (x^{2} (2 y y') + y^{2} (2 x)) + \frac{d}{d x} [16 y^{4}]$

Step 2.3.6

Move $2$ to the left of $x^{2}$ .

$64 x^{3} + 32 (2 \cdot x^{2} y y' + y^{2} (2 x)) + \frac{d}{d x} [16 y^{4}]$

Step 2.3.7

Move $2$ to the left of $y^{2}$ .

$64 x^{3} + 32 (2 x^{2} y y' + 2 y^{2} x) + \frac{d}{d x} [16 y^{4}]$

Step 2.4

Evaluate $\frac{d}{d x} [16 y^{4}]$ .

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Step 2.4.1

Since $16$ is constant with respect to $x$ , the derivative of $16 y^{4}$ with respect to $x$ is $16 \frac{d}{d x} [y^{4}]$ .

$64 x^{3} + 32 (2 x^{2} y y' + 2 y^{2} x) + 16 \frac{d}{d x} [y^{4}]$

Step 2.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = y$ .

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Step 2.4.2.1

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

$64 x^{3} + 32 (2 x^{2} y y' + 2 y^{2} x) + 16 (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [y])$

Step 2.4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$64 x^{3} + 32 (2 x^{2} y y' + 2 y^{2} x) + 16 (4 {u_{2}}^{3} \frac{d}{d x} [y])$

Step 2.4.2.3

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

$64 x^{3} + 32 (2 x^{2} y y' + 2 y^{2} x) + 16 (4 y^{3} \frac{d}{d x} [y])$

Step 2.4.3

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

$64 x^{3} + 32 (2 x^{2} y y' + 2 y^{2} x) + 16 (4 y^{3} y')$

Step 2.4.4

Multiply $4$ by $16$ .

$64 x^{3} + 32 (2 x^{2} y y' + 2 y^{2} x) + 64 y^{3} y'$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$64 x^{3} + 32 (2 x^{2} y y') + 32 (2 y^{2} x) + 64 y^{3} y'$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Multiply $2$ by $32$ .

$64 x^{3} + 64 (x^{2} y y') + 32 (2 y^{2} x) + 64 y^{3} y'$

Step 2.5.2.2

Multiply $2$ by $32$ .

$64 x^{3} + 64 x^{2} y y' + 64 y^{2} x + 64 y^{3} y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $144 x^{2} - 144 y^{2}$ with respect to $x$ is $\frac{d}{d x} [144 x^{2}] + \frac{d}{d x} [- 144 y^{2}]$ .

$\frac{d}{d x} [144 x^{2}] + \frac{d}{d x} [- 144 y^{2}]$

Step 3.2

Evaluate $\frac{d}{d x} [144 x^{2}]$ .

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Step 3.2.1

Since $144$ is constant with respect to $x$ , the derivative of $144 x^{2}$ with respect to $x$ is $144 \frac{d}{d x} [x^{2}]$ .

$144 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 144 y^{2}]$

Step 3.2.2

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

$144 (2 x) + \frac{d}{d x} [- 144 y^{2}]$

Step 3.2.3

Multiply $2$ by $144$ .

$288 x + \frac{d}{d x} [- 144 y^{2}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 144 y^{2}]$ .

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Step 3.3.1

Since $- 144$ is constant with respect to $x$ , the derivative of $- 144 y^{2}$ with respect to $x$ is $- 144 \frac{d}{d x} [y^{2}]$ .

$288 x - 144 \frac{d}{d x} [y^{2}]$

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 $f (x) = x^{2}$ and $g (x) = y$ .

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Step 3.3.2.1

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

$288 x - 144 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

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 $n = 2$ .

$288 x - 144 (2 u \frac{d}{d x} [y])$

Step 3.3.2.3

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

$288 x - 144 (2 y \frac{d}{d x} [y])$

Step 3.3.3

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

$288 x - 144 (2 y y')$

Step 3.3.4

Multiply $2$ by $- 144$ .

$288 x - 288 y y'$

Step 3.4

Reorder terms.

$- 288 y y' + 288 x$

Step 4

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

$64 x^{3} + 64 x^{2} y y' + 64 y^{2} x + 64 y^{3} y' = - 288 y y' + 288 x$

Step 5

Solve for $y'$ .

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Step 5.1

Add $288 y y'$ to both sides of the equation.

$64 x^{3} + 64 x^{2} y y' + 64 y^{2} x + 64 y^{3} y' + 288 y y' = 288 x$

Step 5.2

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

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Step 5.2.1

Subtract $64 x^{3}$ from both sides of the equation.

$64 x^{2} y y' + 64 y^{2} x + 64 y^{3} y' + 288 y y' = 288 x - 64 x^{3}$

Step 5.2.2

Subtract $64 y^{2} x$ from both sides of the equation.

$64 x^{2} y y' + 64 y^{3} y' + 288 y y' = 288 x - 64 x^{3} - 64 y^{2} x$

Step 5.3

Factor $32 y y'$ out of $64 x^{2} y y' + 64 y^{3} y' + 288 y y'$ .

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Step 5.3.1

Factor $32 y y'$ out of $64 x^{2} y y'$ .

$32 y y' (2 x^{2}) + 64 y^{3} y' + 288 y y' = 288 x - 64 x^{3} - 64 y^{2} x$

Step 5.3.2

Factor $32 y y'$ out of $64 y^{3} y'$ .

$32 y y' (2 x^{2}) + 32 y y' (2 y^{2}) + 288 y y' = 288 x - 64 x^{3} - 64 y^{2} x$

Step 5.3.3

Factor $32 y y'$ out of $288 y y'$ .

$32 y y' (2 x^{2}) + 32 y y' (2 y^{2}) + 32 y y' \cdot 9 = 288 x - 64 x^{3} - 64 y^{2} x$

Step 5.3.4

Factor $32 y y'$ out of $32 y y' (2 x^{2}) + 32 y y' (2 y^{2})$ .

$32 y y' (2 x^{2} + 2 y^{2}) + 32 y y' \cdot 9 = 288 x - 64 x^{3} - 64 y^{2} x$

Step 5.3.5

Factor $32 y y'$ out of $32 y y' (2 x^{2} + 2 y^{2}) + 32 y y' \cdot 9$ .

$32 y y' (2 x^{2} + 2 y^{2} + 9) = 288 x - 64 x^{3} - 64 y^{2} x$

Step 5.4

Divide each term in $32 y y' (2 x^{2} + 2 y^{2} + 9) = 288 x - 64 x^{3} - 64 y^{2} x$ by $32 y (2 x^{2} + 2 y^{2} + 9)$ and simplify.

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Step 5.4.1

Divide each term in $32 y y' (2 x^{2} + 2 y^{2} + 9) = 288 x - 64 x^{3} - 64 y^{2} x$ by $32 y (2 x^{2} + 2 y^{2} + 9)$ .

$\frac{32 y y' (2 x^{2} + 2 y^{2} + 9)}{32 y (2 x^{2} + 2 y^{2} + 9)} = \frac{288 x}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $32$ .

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Step 5.4.2.1.1

Cancel the common factor.

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y y' (2 x^{2} + 2 y^{2} + 9)}{y (2 x^{2} + 2 y^{2} + 9)} = \frac{288 x}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.2.2

Cancel the common factor of $y$ .

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Step 5.4.2.2.1

Cancel the common factor.

Step 5.4.2.2.2

Rewrite the expression.

$\frac{y' (2 x^{2} + 2 y^{2} + 9)}{2 x^{2} + 2 y^{2} + 9} = \frac{288 x}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.2.3

Cancel the common factor of $2 x^{2} + 2 y^{2} + 9$ .

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Step 5.4.2.3.1

Cancel the common factor.

Step 5.4.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{288 x}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $288$ and $32$ .

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Step 5.4.3.1.1.1

Factor $32$ out of $288 x$ .

$y' = \frac{32 (9 x)}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Factor $32$ out of $32 y (2 x^{2} + 2 y^{2} + 9)$ .

$y' = \frac{32 (9 x)}{32 (y (2 x^{2} + 2 y^{2} + 9))} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.1.2.2

Cancel the common factor.

$y' = \frac{32 (9 x)}{32 (y (2 x^{2} + 2 y^{2} + 9))} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.1.2.3

Rewrite the expression.

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 x^{3}}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.2

Cancel the common factor of $- 64$ and $32$ .

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Step 5.4.3.1.2.1

Factor $32$ out of $- 64 x^{3}$ .

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{32 (- 2 x^{3})}{32 y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.2.2

Cancel the common factors.

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Step 5.4.3.1.2.2.1

Factor $32$ out of $32 y (2 x^{2} + 2 y^{2} + 9)$ .

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{32 (- 2 x^{3})}{32 (y (2 x^{2} + 2 y^{2} + 9))} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.2.2.2

Cancel the common factor.

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{32 (- 2 x^{3})}{32 (y (2 x^{2} + 2 y^{2} + 9))} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.2.2.3

Rewrite the expression.

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 2 x^{3}}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.3

Move the negative in front of the fraction.

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} - \frac{2 x^{3}}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{- 64 y^{2} x}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.4

Cancel the common factor of $- 64$ and $32$ .

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Step 5.4.3.1.4.1

Factor $32$ out of $- 64 y^{2} x$ .

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} - \frac{2 x^{3}}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{32 (- 2 y^{2} x)}{32 y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.1.4.2

Cancel the common factors.

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Step 5.4.3.1.4.2.1

Factor $32$ out of $32 y (2 x^{2} + 2 y^{2} + 9)$ .

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} - \frac{2 x^{3}}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{32 (- 2 y^{2} x)}{32 (y (2 x^{2} + 2 y^{2} + 9))}$

Step 5.4.3.1.4.2.2

Cancel the common factor.

$y' = \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} - \frac{2 x^{3}}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{32 (- 2 y^{2} x)}{32 (y (2 x^{2} + 2 y^{2} + 9))}$

Step 5.4.3.1.4.2.3

Rewrite the expression.

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

Step 5.4.3.1.5

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

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Step 5.4.3.1.5.1

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

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

Step 5.4.3.1.5.2

Cancel the common factors.

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Step 5.4.3.1.5.2.1

Cancel the common factor.

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

Step 5.4.3.1.5.2.2

Rewrite the expression.

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

Step 5.4.3.1.6

Move the negative in front of the fraction.

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

Step 5.4.3.2

To write $- \frac{2 y x}{2 x^{2} + 2 y^{2} + 9}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$y' = - \frac{2 x^{3}}{y (2 x^{2} + 2 y^{2} + 9)} + \frac{9 x}{y (2 x^{2} + 2 y^{2} + 9)} - \frac{2 y x}{2 x^{2} + 2 y^{2} + 9} \cdot \frac{y}{y}$

Step 5.4.3.3

Write each expression with a common denominator of $y (2 x^{2} + 2 y^{2} + 9)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.1

Multiply $\frac{2 y x}{2 x^{2} + 2 y^{2} + 9}$ by $\frac{y}{y}$ .

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

Step 5.4.3.3.2

Reorder the factors of $(2 x^{2} + 2 y^{2} + 9) y$ .

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

Step 5.4.3.4

Combine the numerators over the common denominator.

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

Step 5.4.3.5

Combine the numerators over the common denominator.

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

Step 5.4.3.6

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

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Step 5.4.3.6.1

Move $y$ .

$y' = \frac{- 2 x^{3} + 9 x - 2 (y \cdot y) x}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.6.2

Multiply $y$ by $y$ .

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

Step 5.4.3.7

Factor $x$ out of $- 2 x^{3} + 9 x - 2 y^{2} x$ .

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Step 5.4.3.7.1

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

$y' = \frac{x (- 2 x^{2}) + 9 x - 2 y^{2} x}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.7.2

Factor $x$ out of $9 x$ .

$y' = \frac{x (- 2 x^{2}) + x \cdot 9 - 2 y^{2} x}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.7.3

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

$y' = \frac{x (- 2 x^{2}) + x \cdot 9 + x (- 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.7.4

Factor $x$ out of $x (- 2 x^{2}) + x \cdot 9$ .

$y' = \frac{x (- 2 x^{2} + 9) + x (- 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.7.5

Factor $x$ out of $x (- 2 x^{2} + 9) + x (- 2 y^{2})$ .

$y' = \frac{x (- 2 x^{2} + 9 - 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.8

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

$y' = \frac{x (- (2 x^{2}) + 9 - 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.9

Rewrite $9$ as $- 1 (- 9)$ .

$y' = \frac{x (- (2 x^{2}) - 1 (- 9) - 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.10

Factor $- 1$ out of $- (2 x^{2}) - 1 (- 9)$ .

$y' = \frac{x (- (2 x^{2} - 9) - 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.11

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

$y' = \frac{x (- (2 x^{2} - 9) - (2 y^{2}))}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.12

Factor $- 1$ out of $- (2 x^{2} - 9) - (2 y^{2})$ .

$y' = \frac{x (- (2 x^{2} - 9 + 2 y^{2}))}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.13

Simplify the expression.

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Step 5.4.3.13.1

Rewrite $- (2 x^{2} - 9 + 2 y^{2})$ as $- 1 (2 x^{2} - 9 + 2 y^{2})$ .

$y' = \frac{x (- 1 (2 x^{2} - 9 + 2 y^{2}))}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 5.4.3.13.2

Move the negative in front of the fraction.

$y' = - \frac{x (2 x^{2} - 9 + 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$

Step 6

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

$\frac{d y}{d x} = - \frac{x (2 x^{2} - 9 + 2 y^{2})}{y (2 x^{2} + 2 y^{2} + 9)}$