Calculus Examples

$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\frac{x^{2}}{16} + \frac{y^{2}}{9}) = \frac{d}{d x} (1)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $\frac{x^{2}}{16} + \frac{y^{2}}{9}$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{2}}{16}] + \frac{d}{d x} [\frac{y^{2}}{9}]$ .

$\frac{d}{d x} [\frac{x^{2}}{16}] + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{x^{2}}{16}]$ .

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Step 2.2.1

Since $\frac{1}{16}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{16}$ with respect to $x$ is $\frac{1}{16} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{16} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2.2

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

$\frac{1}{16} (2 x) + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2.3

Combine $2$ and $\frac{1}{16}$ .

$\frac{2}{16} x + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2.4

Combine $\frac{2}{16}$ and $x$ .

$\frac{2 x}{16} + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2.5

Cancel the common factor of $2$ and $16$ .

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Step 2.2.5.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{16} + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2.5.2

Cancel the common factors.

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Step 2.2.5.2.1

Factor $2$ out of $16$ .

$\frac{2 x}{2 \cdot 8} + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2.5.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 8} + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.2.5.2.3

Rewrite the expression.

$\frac{x}{8} + \frac{d}{d x} [\frac{y^{2}}{9}]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{y^{2}}{9}]$ .

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Step 2.3.1

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{y^{2}}{9}$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [y^{2}]$ .

$\frac{x}{8} + \frac{1}{9} \frac{d}{d x} [y^{2}]$

Step 2.3.2

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 2.3.2.1

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

$\frac{x}{8} + \frac{1}{9} (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.3.2.2

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

$\frac{x}{8} + \frac{1}{9} (2 u \frac{d}{d x} [y])$

Step 2.3.2.3

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

$\frac{x}{8} + \frac{1}{9} (2 y \frac{d}{d x} [y])$

Step 2.3.3

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

$\frac{x}{8} + \frac{1}{9} (2 y y')$

Step 2.3.4

Combine $2$ and $\frac{1}{9}$ .

$\frac{x}{8} + \frac{2}{9} (y y')$

Step 2.3.5

Combine $y$ and $\frac{2}{9}$ .

$\frac{x}{8} + \frac{y \cdot 2}{9} y'$

Step 2.3.6

Combine $\frac{y \cdot 2}{9}$ and $y'$ .

$\frac{x}{8} + \frac{y \cdot 2 y'}{9}$

Step 2.3.7

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

$\frac{x}{8} + \frac{2 y y'}{9}$

Step 3

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

$0$

Step 4

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

$\frac{x}{8} + \frac{2 y y'}{9} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $\frac{x}{8}$ from both sides of the equation.

$\frac{2 y y'}{9} = - \frac{x}{8}$

Step 5.2

Multiply both sides of the equation by $\frac{9}{2}$ .

$\frac{9}{2} \cdot \frac{2 y y'}{9} = \frac{9}{2} (- \frac{x}{8})$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{9}{2} \cdot \frac{2 y y'}{9}$ .

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Step 5.3.1.1.1

Cancel the common factor of $9$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{9}{2} \cdot \frac{2 y y'}{9} = \frac{9}{2} (- \frac{x}{8})$

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 y y') = \frac{9}{2} (- \frac{x}{8})$

Step 5.3.1.1.2

Cancel the common factor of $2$ .

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Step 5.3.1.1.2.1

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

$\frac{1}{2} (2 (y y')) = \frac{9}{2} (- \frac{x}{8})$

Step 5.3.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 (y y')) = \frac{9}{2} (- \frac{x}{8})$

Step 5.3.1.1.2.3

Rewrite the expression.

$y y' = \frac{9}{2} (- \frac{x}{8})$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $\frac{9}{2} (- \frac{x}{8})$ .

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Step 5.3.2.1.1

Multiply $\frac{9}{2}$ by $\frac{x}{8}$ .

$y y' = - \frac{9 x}{2 \cdot 8}$

Step 5.3.2.1.2

Multiply $2$ by $8$ .

$y y' = - \frac{9 x}{16}$

Step 5.4

Divide each term in $y y' = - \frac{9 x}{16}$ by $y$ and simplify.

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Step 5.4.1

Divide each term in $y y' = - \frac{9 x}{16}$ by $y$ .

$\frac{y y'}{y} = \frac{- \frac{9 x}{16}}{y}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $y$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{- \frac{9 x}{16}}{y}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{9 x}{16}}{y}$

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.

$y' = - \frac{9 x}{16} \cdot \frac{1}{y}$

Step 5.4.3.2

Multiply $\frac{1}{y}$ by $\frac{9 x}{16}$ .

$y' = - \frac{9 x}{y \cdot 16}$

Step 5.4.3.3

Move $16$ to the left of $y$ .

$y' = - \frac{9 x}{16 y}$

Step 6

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

$\frac{d y}{d x} = - \frac{9 x}{16 y}$