Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = \frac{y}{x}$ .

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

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 $f (x) = y$ and $g (x) = x$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Multiply $- 1$ by $1$ .

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

Step 3.6

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

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

Step 3.7

Simplify the expression.

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Step 3.7.1

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

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

Step 3.7.2

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

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Simplify the right side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor.

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

Step 5.2.1.1.3

Rewrite the expression.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Multiply $- - y$ .

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Step 5.2.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.1.3.2

Multiply $y$ by $1$ .

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

Step 5.2.1.4

Move $x$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Rewrite $- 1 y'$ as $- y'$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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Step 5.3.3.1

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

$y' x \cdot x + y' x = y$

Step 5.3.3.2

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

$y' x \cdot x + y' x \cdot 1 = y$

Step 5.3.3.3

Factor $y' x$ out of $y' x \cdot x + y' x \cdot 1$ .

$y' x (x + 1) = y$

Step 5.3.4

Divide each term in $y' x (x + 1) = y$ by $x (x + 1)$ and simplify.

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Step 5.3.4.1

Divide each term in $y' x (x + 1) = y$ by $x (x + 1)$ .

$\frac{y' x (x + 1)}{x (x + 1)} = \frac{y}{x (x + 1)}$

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

Cancel the common factor of $x$ .

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Step 5.3.4.2.1.1

Cancel the common factor.

$\frac{y' x (x + 1)}{x (x + 1)} = \frac{y}{x (x + 1)}$

Step 5.3.4.2.1.2

Rewrite the expression.

$\frac{y' (x + 1)}{x + 1} = \frac{y}{x (x + 1)}$

Step 5.3.4.2.2

Cancel the common factor of $x + 1$ .

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Step 5.3.4.2.2.1

Cancel the common factor.

$\frac{y' (x + 1)}{x + 1} = \frac{y}{x (x + 1)}$

Step 5.3.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{y}{x (x + 1)}$

Step 6

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

$\frac{d y}{d x} = \frac{y}{x (x + 1)}$