Calculus Examples

$y = \frac{2 x - 4 y}{4 x - 2 y}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{2 x - 4 y}{4 x - 2 y})$

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

Cancel the common factor of $2 x - 4 y$ and $4 x - 2 y$ .

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Step 3.1.1

Factor $2$ out of $2 x$ .

$\frac{d}{d x} [\frac{2 (x) - 4 y}{4 x - 2 y}]$

Step 3.1.2

Factor $2$ out of $- 4 y$ .

$\frac{d}{d x} [\frac{2 (x) + 2 (- 2 y)}{4 x - 2 y}]$

Step 3.1.3

Factor $2$ out of $2 (x) + 2 (- 2 y)$ .

$\frac{d}{d x} [\frac{2 (x - 2 y)}{4 x - 2 y}]$

Step 3.1.4

Cancel the common factors.

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Step 3.1.4.1

Factor $2$ out of $4 x$ .

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

Step 3.1.4.2

Factor $2$ out of $- 2 y$ .

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

Step 3.1.4.3

Factor $2$ out of $2 (2 x) + 2 (- y)$ .

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

Step 3.1.4.4

Cancel the common factor.

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

Step 3.1.4.5

Rewrite the expression.

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

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 $f (x) = x - 2 y$ and $g (x) = 2 x - y$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Multiply $2$ by $1$ .

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

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

Step 3.11.2

Simplify the numerator.

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Step 3.11.2.1

Simplify each term.

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Step 3.11.2.1.1

Expand $(2 x - y) (1 - 2 y')$ using the FOIL Method.

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Step 3.11.2.1.1.1

Apply the distributive property.

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

Step 3.11.2.1.1.2

Apply the distributive property.

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

Step 3.11.2.1.1.3

Apply the distributive property.

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

Step 3.11.2.1.2

Simplify each term.

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Step 3.11.2.1.2.1

Multiply $2$ by $1$ .

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

Step 3.11.2.1.2.2

Multiply $- 2$ by $2$ .

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

Step 3.11.2.1.2.3

Multiply $- 1$ by $1$ .

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

Step 3.11.2.1.2.4

Multiply $- 2$ by $- 1$ .

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

Step 3.11.2.1.3

Multiply $- 2$ by $- 1$ .

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

Step 3.11.2.1.4

Expand $(- x + 2 y) (2 - y')$ using the FOIL Method.

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Step 3.11.2.1.4.1

Apply the distributive property.

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

Step 3.11.2.1.4.2

Apply the distributive property.

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

Step 3.11.2.1.4.3

Apply the distributive property.

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

Step 3.11.2.1.5

Simplify each term.

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Step 3.11.2.1.5.1

Multiply $2$ by $- 1$ .

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

Step 3.11.2.1.5.2

Multiply $- x (- y')$ .

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Step 3.11.2.1.5.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.11.2.1.5.2.2

Multiply $x$ by $1$ .

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

Step 3.11.2.1.5.3

Multiply $2$ by $2$ .

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

Step 3.11.2.1.5.4

Multiply $- 1$ by $2$ .

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

Step 3.11.2.2

Combine the opposite terms in $2 x - 4 x y' - y + 2 y y' - 2 x + x y' + 4 y - 2 y y'$ .

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Step 3.11.2.2.1

Subtract $2 x$ from $2 x$ .

$\frac{- 4 x y' - y + 2 y y' + 0 + x y' + 4 y - 2 y y'}{{(2 x - y)}^{2}}$

Step 3.11.2.2.2

Add $- 4 x y' - y + 2 y y'$ and $0$ .

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

Step 3.11.2.2.3

Subtract $2 y y'$ from $2 y y'$ .

$\frac{- 4 x y' - y + x y' + 4 y + 0}{{(2 x - y)}^{2}}$

Step 3.11.2.2.4

Add $- 4 x y' - y + x y' + 4 y$ and $0$ .

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

Step 3.11.2.3

Add $- 4 x y'$ and $x y'$ .

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

Step 3.11.2.4

Add $- y$ and $4 y$ .

$\frac{- 3 x y' + 3 y}{{(2 x - y)}^{2}}$

Step 3.11.3

Factor $3$ out of $- 3 x y' + 3 y$ .

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Step 3.11.3.1

Factor $3$ out of $- 3 x y'$ .

$\frac{3 (- x y') + 3 y}{{(2 x - y)}^{2}}$

Step 3.11.3.2

Factor $3$ out of $3 y$ .

$\frac{3 (- x y') + 3 (y)}{{(2 x - y)}^{2}}$

Step 3.11.3.3

Factor $3$ out of $3 (- x y') + 3 (y)$ .

$\frac{3 (- x y' + y)}{{(2 x - y)}^{2}}$

Step 3.11.4

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

$\frac{3 (- (x y') + y)}{{(2 x - y)}^{2}}$

Step 3.11.5

Factor $- 1$ out of $y$ .

$\frac{3 (- (x y') - 1 (- y))}{{(2 x - y)}^{2}}$

Step 3.11.6

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

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

Step 3.11.7

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

$\frac{3 (- 1 (x y' - y))}{{(2 x - y)}^{2}}$

Step 3.11.8

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{3 (x y' - y)}{{(2 x - y)}^{2}} {(2 x - y)}^{2}$ .

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Move the leading negative in $- \frac{3 (x y' - y)}{{(2 x - y)}^{2}}$ into the numerator.

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

Step 5.2.1.1.2

Cancel the common factor.

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

Step 5.2.1.1.3

Rewrite the expression.

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

Step 5.2.1.2

Apply the distributive property.

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

Step 5.2.1.3

Simplify the expression.

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Step 5.2.1.3.1

Multiply $- 1$ by $- 3$ .

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

Step 5.2.1.3.2

Move $x$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Move all terms containing $y'$ to the left side of the equation.

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Step 5.3.1.1

Add $3 y' x$ to both sides of the equation.

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

Step 5.3.1.2

Simplify each term.

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Step 5.3.1.2.1

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

$y' ((2 x - y) (2 x - y)) + 3 y' x = 3 y$

Step 5.3.1.2.2

Expand $(2 x - y) (2 x - y)$ using the FOIL Method.

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Step 5.3.1.2.2.1

Apply the distributive property.

$y' (2 x (2 x - y) - y (2 x - y)) + 3 y' x = 3 y$

Step 5.3.1.2.2.2

Apply the distributive property.

$y' (2 x (2 x) + 2 x (- y) - y (2 x - y)) + 3 y' x = 3 y$

Step 5.3.1.2.2.3

Apply the distributive property.

$y' (2 x (2 x) + 2 x (- y) - y (2 x) - y (- y)) + 3 y' x = 3 y$

Step 5.3.1.2.3

Simplify and combine like terms.

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Step 5.3.1.2.3.1

Simplify each term.

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Step 5.3.1.2.3.1.1

Rewrite using the commutative property of multiplication.

$y' (2 \cdot 2 x \cdot x + 2 x (- y) - y (2 x) - y (- y)) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.2

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

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Step 5.3.1.2.3.1.2.1

Move $x$ .

$y' (2 \cdot 2 (x \cdot x) + 2 x (- y) - y (2 x) - y (- y)) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.2.2

Multiply $x$ by $x$ .

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

Step 5.3.1.2.3.1.3

Multiply $2$ by $2$ .

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

Step 5.3.1.2.3.1.4

Rewrite using the commutative property of multiplication.

$y' (4 x^{2} + 2 \cdot - 1 x y - y (2 x) - y (- y)) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.5

Multiply $2$ by $- 1$ .

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

Step 5.3.1.2.3.1.6

Rewrite using the commutative property of multiplication.

$y' (4 x^{2} - 2 x y - 1 \cdot 2 y x - y (- y)) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.7

Multiply $- 1$ by $2$ .

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

Step 5.3.1.2.3.1.8

Rewrite using the commutative property of multiplication.

$y' (4 x^{2} - 2 x y - 2 y x - 1 \cdot - 1 y \cdot y) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.9

Multiply $y$ by $y$ by adding the exponents.

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Step 5.3.1.2.3.1.9.1

Move $y$ .

$y' (4 x^{2} - 2 x y - 2 y x - 1 \cdot - 1 (y \cdot y)) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.9.2

Multiply $y$ by $y$ .

$y' (4 x^{2} - 2 x y - 2 y x - 1 \cdot - 1 y^{2}) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.10

Multiply $- 1$ by $- 1$ .

$y' (4 x^{2} - 2 x y - 2 y x + 1 y^{2}) + 3 y' x = 3 y$

Step 5.3.1.2.3.1.11

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

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

Step 5.3.1.2.3.2

Subtract $2 y x$ from $- 2 x y$ .

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Step 5.3.1.2.3.2.1

Move $y$ .

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

Step 5.3.1.2.3.2.2

Subtract $2 x y$ from $- 2 x y$ .

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

Step 5.3.1.2.4

Apply the distributive property.

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

Step 5.3.1.2.5

Simplify.

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Step 5.3.1.2.5.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.2.5.2

Rewrite using the commutative property of multiplication.

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

Step 5.3.2

Factor $y'$ out of $4 y' x^{2} - 4 y' x y + y' y^{2} + 3 y' x$ .

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Step 5.3.2.1

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

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

Step 5.3.2.2

Factor $y'$ out of $- 4 y' x y$ .

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

Step 5.3.2.3

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

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

Step 5.3.2.4

Factor $y'$ out of $3 y' x$ .

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

Step 5.3.2.5

Factor $y'$ out of $y' (4 x^{2}) + y' (- 4 x y)$ .

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

Step 5.3.2.6

Factor $y'$ out of $y' (4 x^{2} - 4 x y) + y' (y^{2})$ .

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

Step 5.3.2.7

Factor $y'$ out of $y' (4 x^{2} - 4 x y + y^{2}) + y' (3 x)$ .

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

Step 5.3.3

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

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Step 5.3.3.1

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

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

Step 5.3.3.2

Simplify the left side.

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Step 5.3.3.2.1

Cancel the common factor of $4 x^{2} - 4 x y + y^{2} + 3 x$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $y'$ by $1$ .

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

Step 6

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

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