Calculus Examples

$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}}) = \frac{d}{d y} (1)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

$\frac{1}{a^{2}} \frac{d}{d y} [x^{2}] + \frac{d}{d y} [- \frac{y^{2}}{b^{2}}]$

Step 2.2.2

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

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Step 2.2.2.1

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

$\frac{1}{a^{2}} (\frac{d}{d u} [u^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [- \frac{y^{2}}{b^{2}}]$

Step 2.2.2.2

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

$\frac{1}{a^{2}} (2 u \frac{d}{d y} [x]) + \frac{d}{d y} [- \frac{y^{2}}{b^{2}}]$

Step 2.2.2.3

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

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

Step 2.2.3

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

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

Step 2.2.4

Combine $2$ and $\frac{1}{a^{2}}$ .

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

Step 2.2.5

Combine $x$ and $\frac{2}{a^{2}}$ .

$\frac{x \cdot 2}{a^{2}} x' + \frac{d}{d y} [- \frac{y^{2}}{b^{2}}]$

Step 2.2.6

Combine $\frac{x \cdot 2}{a^{2}}$ and $x'$ .

$\frac{x \cdot 2 x'}{a^{2}} + \frac{d}{d y} [- \frac{y^{2}}{b^{2}}]$

Step 2.2.7

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

$\frac{2 x x'}{a^{2}} + \frac{d}{d y} [- \frac{y^{2}}{b^{2}}]$

Step 2.3

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

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Step 2.3.1

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

$\frac{2 x x'}{a^{2}} - \frac{1}{b^{2}} \frac{d}{d y} [y^{2}]$

Step 2.3.2

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

$\frac{2 x x'}{a^{2}} - \frac{1}{b^{2}} (2 y)$

Step 2.3.3

Multiply $2$ by $- 1$ .

$\frac{2 x x'}{a^{2}} - 2 \frac{1}{b^{2}} y$

Step 2.3.4

Combine $- 2$ and $\frac{1}{b^{2}}$ .

$\frac{2 x x'}{a^{2}} + \frac{- 2}{b^{2}} y$

Step 2.3.5

Combine $\frac{- 2}{b^{2}}$ and $y$ .

$\frac{2 x x'}{a^{2}} + \frac{- 2 y}{b^{2}}$

Step 2.3.6

Move the negative in front of the fraction.

$\frac{2 x x'}{a^{2}} - \frac{2 y}{b^{2}}$

Step 3

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$0$

Step 4

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

$\frac{2 x x'}{a^{2}} - \frac{2 y}{b^{2}} = 0$

Step 5

Solve for $x'$ .

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Step 5.1

Add $\frac{2 y}{b^{2}}$ to both sides of the equation.

$\frac{2 x x'}{a^{2}} = \frac{2 y}{b^{2}}$

Step 5.2

Multiply both sides by $a^{2}$ .

$\frac{2 x x'}{a^{2}} a^{2} = \frac{2 y}{b^{2}} a^{2}$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $a^{2}$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{2 x x'}{a^{2}} a^{2} = \frac{2 y}{b^{2}} a^{2}$

Step 5.3.1.1.2

Rewrite the expression.

$2 x x' = \frac{2 y}{b^{2}} a^{2}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Combine $\frac{2 y}{b^{2}}$ and $a^{2}$ .

$2 x x' = \frac{2 y a^{2}}{b^{2}}$

Step 5.4

Divide each term in $2 x x' = \frac{2 y a^{2}}{b^{2}}$ by $2 x$ and simplify.

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Step 5.4.1

Divide each term in $2 x x' = \frac{2 y a^{2}}{b^{2}}$ by $2 x$ .

$\frac{2 x x'}{2 x} = \frac{\frac{2 y a^{2}}{b^{2}}}{2 x}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 x x'}{2 x} = \frac{\frac{2 y a^{2}}{b^{2}}}{2 x}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{x x'}{x} = \frac{\frac{2 y a^{2}}{b^{2}}}{2 x}$

Step 5.4.2.2

Cancel the common factor of $x$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{x x'}{x} = \frac{\frac{2 y a^{2}}{b^{2}}}{2 x}$

Step 5.4.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{\frac{2 y a^{2}}{b^{2}}}{2 x}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x' = \frac{2 y a^{2}}{b^{2}} \cdot \frac{1}{2 x}$

Step 5.4.3.2

Combine.

$x' = \frac{2 y a^{2} \cdot 1}{b^{2} (2 x)}$

Step 5.4.3.3

Cancel the common factor of $2$ .

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Step 5.4.3.3.1

Cancel the common factor.

$x' = \frac{2 y a^{2} \cdot 1}{b^{2} (2 x)}$

Step 5.4.3.3.2

Rewrite the expression.

$x' = \frac{y a^{2} \cdot 1}{b^{2} (x)}$

Step 5.4.3.4

Multiply $y$ by $1$ .

$x' = \frac{a^{2} y}{b^{2} x}$

Step 6

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

$\frac{d x}{d y} = \frac{a^{2} y}{b^{2} x}$