Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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 $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $y$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $2$ by $- 1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Combine terms.

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Step 2.4.1.1

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{y^{2}}{y^{2}}$ .

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

Step 2.4.1.2

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

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

Step 2.4.1.3

Combine the numerators over the common denominator.

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

Step 2.4.2

Reorder terms.

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

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.

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

Step 5

Solve for $y'$ .

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Step 5.1

Set the numerator equal to zero.

$- x y' - 2 y^{2} x + y = 0$

Step 5.2

Solve the equation for $y'$ .

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Step 5.2.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.2.1.1

Add $2 y^{2} x$ to both sides of the equation.

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

Step 5.2.1.2

Subtract $y$ from both sides of the equation.

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

Step 5.2.2

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

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Step 5.2.2.1

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

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

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.

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

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.

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

Step 5.2.2.2.2.2

Divide $y'$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Simplify each term.

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Step 5.2.2.3.1.1

Cancel the common factor of $x$ .

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Step 5.2.2.3.1.1.1

Cancel the common factor.

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

Step 5.2.2.3.1.1.2

Rewrite the expression.

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

Step 5.2.2.3.1.1.3

Move the negative one from the denominator of $\frac{2 y^{2}}{- 1}$ .

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

Step 5.2.2.3.1.2

Rewrite $- 1 \cdot (2 y^{2})$ as $- (2 y^{2})$ .

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

Step 5.2.2.3.1.3

Multiply $2$ by $- 1$ .

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

Step 5.2.2.3.1.4

Dividing two negative values results in a positive value.

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

Step 6

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

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