Calculus Examples

$15 (x^{2} + y^{2}) = 289 (x^{2} - y^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (15 (x^{2} + y^{2})) = \frac{d}{d x} (289 (x^{2} - y^{2}))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

Since $15$ is constant with respect to $x$ , the derivative of $15 (x^{2} + y^{2})$ with respect to $x$ is $15 \frac{d}{d x} [x^{2} + y^{2}]$ .

$15 \frac{d}{d x} [x^{2} + y^{2}]$

Step 2.1.2

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

$15 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

Step 2.1.3

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

$15 (2 x + \frac{d}{d x} [y^{2}])$

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 $g (x) = y$ .

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Step 2.2.1

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

$15 (2 x + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.2.2

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

$15 (2 x + 2 u \frac{d}{d x} [y])$

Step 2.2.3

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

$15 (2 x + 2 y \frac{d}{d x} [y])$

Step 2.3

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

$15 (2 x + 2 y y')$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$15 (2 x) + 15 (2 y y')$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $2$ by $15$ .

$30 x + 15 (2 y y')$

Step 2.4.2.2

Multiply $2$ by $15$ .

$30 x + 30 y y'$

Step 2.4.3

Reorder terms.

$30 y y' + 30 x$

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 $289$ is constant with respect to $x$ , the derivative of $289 (x^{2} - y^{2})$ with respect to $x$ is $289 \frac{d}{d x} [x^{2} - y^{2}]$ .

$289 \frac{d}{d x} [x^{2} - y^{2}]$

Step 3.1.2

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

$289 (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- y^{2}])$

Step 3.1.3

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

$289 (2 x + \frac{d}{d x} [- y^{2}])$

Step 3.1.4

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

$289 (2 x - \frac{d}{d x} [y^{2}])$

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 $y$ .

$289 (2 x - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]))$

Step 3.2.2

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

$289 (2 x - (2 u \frac{d}{d x} [y]))$

Step 3.2.3

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

$289 (2 x - (2 y \frac{d}{d x} [y]))$

Step 3.3

Multiply $2$ by $- 1$ .

$289 (2 x - 2 (y \frac{d}{d x} [y]))$

Step 3.4

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

$289 (2 x - 2 y y')$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$289 (2 x) + 289 (- 2 y y')$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Multiply $2$ by $289$ .

$578 x + 289 (- 2 y y')$

Step 3.5.2.2

Multiply $- 2$ by $289$ .

$578 x - 578 y y'$

Step 3.5.3

Reorder terms.

$- 578 y y' + 578 x$

Step 4

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

$30 y y' + 30 x = - 578 y y' + 578 x$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Add $578 y y'$ to both sides of the equation.

$30 y y' + 30 x + 578 y y' = 578 x$

Step 5.1.2

Add $30 y y'$ and $578 y y'$ .

$608 y y' + 30 x = 578 x$

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 $30 x$ from both sides of the equation.

$608 y y' = 578 x - 30 x$

Step 5.2.2

Subtract $30 x$ from $578 x$ .

$608 y y' = 548 x$

Step 5.3

Divide each term in $608 y y' = 548 x$ by $608 y$ and simplify.

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Step 5.3.1

Divide each term in $608 y y' = 548 x$ by $608 y$ .

$\frac{608 y y'}{608 y} = \frac{548 x}{608 y}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $608$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{608 y y'}{608 y} = \frac{548 x}{608 y}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{548 x}{608 y}$

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{548 x}{608 y}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{548 x}{608 y}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $548$ and $608$ .

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Step 5.3.3.1.1

Factor $4$ out of $548 x$ .

$y' = \frac{4 (137 x)}{608 y}$

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

Factor $4$ out of $608 y$ .

$y' = \frac{4 (137 x)}{4 (152 y)}$

Step 5.3.3.1.2.2

Cancel the common factor.

$y' = \frac{4 (137 x)}{4 (152 y)}$

Step 5.3.3.1.2.3

Rewrite the expression.

$y' = \frac{137 x}{152 y}$

Step 6

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

$\frac{d y}{d x} = \frac{137 x}{152 y}$