Calculus Examples

$y = - \frac{x}{y}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (- \frac{x}{y})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = - 1$ and $g (x) = \frac{x}{y}$ .

$- \frac{d}{d x} [\frac{x}{y}] + \frac{x}{y} \frac{d}{d x} [- 1]$

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

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

Step 3.3.2

Multiply $y$ by $1$ .

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

Step 3.4

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

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

Step 3.5

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

$- \frac{y - x y'}{y^{2}} + \frac{x}{y} \cdot 0$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $\frac{x}{y}$ by $0$ .

$- \frac{y - x y'}{y^{2}} + 0$

Step 3.6.2

Add $- \frac{y - x y'}{y^{2}}$ and $0$ .

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{y - x y'}{y^{2}} y^{2}$ .

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Move the leading negative in $- \frac{y - x y'}{y^{2}}$ into the numerator.

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

Step 5.2.1.1.2

Cancel the common factor.

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

Step 5.2.1.1.3

Rewrite the expression.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Multiply $- (- x y')$ .

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Step 5.2.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.1.3.2

Multiply $x$ by $1$ .

$y' y^{2} = - y + x y'$

Step 5.2.1.4

Reorder $x$ and $y'$ .

$y' y^{2} = - y + y' x$

Step 5.2.1.5

Reorder $- y$ and $y' x$ .

$y' y^{2} = y' x - y$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $y' x$ from both sides of the equation.

$y' y^{2} - y' x = - y$

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

Factor $y'$ out of $- y' x$ .

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

Step 5.3.2.3

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

$y' (y^{2} - 1 x) = - y$

Step 5.3.3

Rewrite $- 1 x$ as $- x$ .

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

Step 5.3.4

Divide each term in $y' (y^{2} - x) = - y$ by $y^{2} - x$ and simplify.

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Step 5.3.4.1

Divide each term in $y' (y^{2} - x) = - y$ by $y^{2} - x$ .

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

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

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

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Step 5.3.4.2.1.1

Cancel the common factor.

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

Step 5.3.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.4.3

Simplify the right side.

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Step 5.3.4.3.1

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{y}{y^{2} - x}$