Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $- 1$ by $1$ .

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Simplify the numerator.

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Step 2.8.3.1

Combine the opposite terms in $y y' - x y' - y y' - y \cdot - 1$ .

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Step 2.8.3.1.1

Subtract $y y'$ from $y y'$ .

$\frac{- x y' + 0 - y \cdot - 1}{{(y - x)}^{2}}$

Step 2.8.3.1.2

Add $- x y'$ and $0$ .

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

Step 2.8.3.2

Multiply $- y \cdot - 1$ .

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Step 2.8.3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.8.3.2.2

Multiply $y$ by $1$ .

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

Step 2.8.4

Reorder terms.

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

Step 2.8.5

Factor $- 1$ out of $- x y'$ .

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

Step 2.8.6

Factor $- 1$ out of $y$ .

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

Step 2.8.7

Factor $- 1$ out of $- (x y') - 1 (- y)$ .

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

Step 2.8.8

Rewrite $- (x y' - y)$ as $- 1 (x y' - y)$ .

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

Step 2.8.9

Move the negative in front of the fraction.

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1]$ .

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

Step 3.2

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

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

Step 3.3

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

$2 x + 0$

Step 3.4

Add $2 x$ and $0$ .

$2 x$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Divide each term in $- \frac{x y' - y}{{(- x + y)}^{2}} = 2 x$ by $- 1$ and simplify.

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Step 5.1.1

Divide each term in $- \frac{x y' - y}{{(- x + y)}^{2}} = 2 x$ by $- 1$ .

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

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Dividing two negative values results in a positive value.

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

Step 5.1.2.2

Divide $\frac{x y' - y}{{(- x + y)}^{2}}$ by $1$ .

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

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

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

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

Step 5.1.3.2

Rewrite $- 1 \cdot (2 x)$ as $- (2 x)$ .

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

Step 5.1.3.3

Multiply $2$ by $- 1$ .

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

Step 5.2

Multiply both sides by ${(- x + y)}^{2}$ .

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

Step 5.3

Simplify the left side.

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Step 5.3.1

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

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Step 5.3.1.1

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

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Step 5.3.1.1.1

Cancel the common factor.

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

Step 5.3.1.1.2

Rewrite the expression.

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

Step 5.3.1.2

Reorder $x$ and $y'$ .

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

Step 5.4

Solve for $y'$ .

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Step 5.4.1

Simplify $- 2 x {(- x + y)}^{2}$ .

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Step 5.4.1.1

Rewrite.

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

Step 5.4.1.2

Rewrite ${(- x + y)}^{2}$ as $(- x + y) (- x + y)$ .

$y' x - y = - 2 x ((- x + y) (- x + y))$

Step 5.4.1.3

Expand $(- x + y) (- x + y)$ using the FOIL Method.

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Step 5.4.1.3.1

Apply the distributive property.

$y' x - y = - 2 x (- x (- x + y) + y (- x + y))$

Step 5.4.1.3.2

Apply the distributive property.

$y' x - y = - 2 x (- x (- x) - x y + y (- x + y))$

Step 5.4.1.3.3

Apply the distributive property.

$y' x - y = - 2 x (- x (- x) - x y + y (- x) + y \cdot y)$

Step 5.4.1.4

Simplify and combine like terms.

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Step 5.4.1.4.1

Simplify each term.

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Step 5.4.1.4.1.1

Rewrite using the commutative property of multiplication.

$y' x - y = - 2 x (- 1 \cdot - 1 x \cdot x - x y + y (- x) + y \cdot y)$

Step 5.4.1.4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.1.4.1.2.1

Move $x$ .

$y' x - y = - 2 x (- 1 \cdot - 1 (x \cdot x) - x y + y (- x) + y \cdot y)$

Step 5.4.1.4.1.2.2

Multiply $x$ by $x$ .

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

Step 5.4.1.4.1.3

Multiply $- 1$ by $- 1$ .

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

Step 5.4.1.4.1.4

Multiply $x^{2}$ by $1$ .

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

Step 5.4.1.4.1.5

Rewrite using the commutative property of multiplication.

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

Step 5.4.1.4.1.6

Multiply $y$ by $y$ .

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

Step 5.4.1.4.2

Subtract $y x$ from $- x y$ .

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Step 5.4.1.4.2.1

Move $y$ .

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

Step 5.4.1.4.2.2

Subtract $x y$ from $- x y$ .

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

Step 5.4.1.5

Apply the distributive property.

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

Step 5.4.1.6

Simplify.

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Step 5.4.1.6.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 5.4.1.6.1.1

Move $x^{2}$ .

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

Step 5.4.1.6.1.2

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

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Step 5.4.1.6.1.2.1

Raise $x$ to the power of $1$ .

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

Step 5.4.1.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.4.1.6.1.3

Add $2$ and $1$ .

$y' x - y = - 2 x^{3} - 2 x (- 2 x y) - 2 x y^{2}$

Step 5.4.1.6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.1.6.2.1

Move $x$ .

$y' x - y = - 2 x^{3} - 2 (x \cdot x) (- 2 y) - 2 x y^{2}$

Step 5.4.1.6.2.2

Multiply $x$ by $x$ .

$y' x - y = - 2 x^{3} - 2 x^{2} (- 2 y) - 2 x y^{2}$

Step 5.4.1.7

Simplify each term.

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Step 5.4.1.7.1

Rewrite using the commutative property of multiplication.

$y' x - y = - 2 x^{3} - 2 \cdot - 2 x^{2} y - 2 x y^{2}$

Step 5.4.1.7.2

Multiply $- 2$ by $- 2$ .

$y' x - y = - 2 x^{3} + 4 x^{2} y - 2 x y^{2}$

Step 5.4.2

Add $y$ to both sides of the equation.

$y' x = - 2 x^{3} + 4 x^{2} y - 2 x y^{2} + y$

Step 5.4.3

Divide each term in $y' x = - 2 x^{3} + 4 x^{2} y - 2 x y^{2} + y$ by $x$ and simplify.

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Step 5.4.3.1

Divide each term in $y' x = - 2 x^{3} + 4 x^{2} y - 2 x y^{2} + y$ by $x$ .

$\frac{y' x}{x} = \frac{- 2 x^{3}}{x} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $x$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{y' x}{x} = \frac{- 2 x^{3}}{x} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x^{3}}{x} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Simplify each term.

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Step 5.4.3.3.1.1

Cancel the common factor of $x^{3}$ and $x$ .

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Step 5.4.3.3.1.1.1

Factor $x$ out of $- 2 x^{3}$ .

$y' = \frac{x (- 2 x^{2})}{x} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.1.2

Cancel the common factors.

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Step 5.4.3.3.1.1.2.1

Raise $x$ to the power of $1$ .

$y' = \frac{x (- 2 x^{2})}{x^{1}} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.1.2.2

Factor $x$ out of $x^{1}$ .

$y' = \frac{x (- 2 x^{2})}{x \cdot 1} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.1.2.3

Cancel the common factor.

$y' = \frac{x (- 2 x^{2})}{x \cdot 1} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.1.2.4

Rewrite the expression.

$y' = \frac{- 2 x^{2}}{1} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.1.2.5

Divide $- 2 x^{2}$ by $1$ .

$y' = - 2 x^{2} + \frac{4 x^{2} y}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.2

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

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Step 5.4.3.3.1.2.1

Factor $x$ out of $4 x^{2} y$ .

$y' = - 2 x^{2} + \frac{x (4 x y)}{x} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.2.2

Cancel the common factors.

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Step 5.4.3.3.1.2.2.1

Raise $x$ to the power of $1$ .

$y' = - 2 x^{2} + \frac{x (4 x y)}{x^{1}} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.2.2.2

Factor $x$ out of $x^{1}$ .

$y' = - 2 x^{2} + \frac{x (4 x y)}{x \cdot 1} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.2.2.3

Cancel the common factor.

$y' = - 2 x^{2} + \frac{x (4 x y)}{x \cdot 1} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.2.2.4

Rewrite the expression.

$y' = - 2 x^{2} + \frac{4 x y}{1} + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.2.2.5

Divide $4 x y$ by $1$ .

$y' = - 2 x^{2} + 4 x y + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.3

Cancel the common factor of $x$ .

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Step 5.4.3.3.1.3.1

Cancel the common factor.

$y' = - 2 x^{2} + 4 x y + \frac{- 2 x y^{2}}{x} + \frac{y}{x}$

Step 5.4.3.3.1.3.2

Divide $- 2 y^{2}$ by $1$ .

$y' = - 2 x^{2} + 4 x y - 2 y^{2} + \frac{y}{x}$

Step 6

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

$\frac{d y}{d x} = - 2 x^{2} + 4 x y - 2 y^{2} + \frac{y}{x}$