Calculus Examples

$- \frac{x}{y} = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- \frac{x}{y}) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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 2.2

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 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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 2.3.2

Multiply $y$ by $1$ .

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

Step 2.4

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

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

Step 2.5

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 2.6

Simplify the expression.

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 3

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

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Set the numerator equal to zero.

$y - x y' = 0$

Step 5.2

Solve the equation for $y'$ .

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Step 5.2.1

Subtract $y$ from both sides of the equation.

$- x y' = - y$

Step 5.2.2

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

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Step 5.2.2.1

Divide each term in $- x y' = - y$ by $- x$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2.2

Cancel the common factor of $x$ .

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Step 5.2.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 6

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

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