Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.4

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

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Step 2.3.1.4.1

Move $y$ .

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

Step 2.3.1.4.2

Multiply $y$ by $y$ .

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

Step 2.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.6

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

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

Step 2.3.2

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

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Step 2.3.2.1

Move $y$ .

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

Step 2.3.2.2

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

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

Step 2.4

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

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

Step 2.5

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = x$ .

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Step 2.5.1

To apply the Chain Rule, set $u$ as $x$ .

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

Step 2.5.2

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

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

Step 2.5.3

Replace all occurrences of $u$ with $x$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = x$ and $g (y) = y$ .

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

Step 2.9

Differentiate using the Power Rule.

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Step 2.9.1

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

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

Step 2.9.2

Multiply $x$ by $1$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Simplify.

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Step 2.12.1

Apply the distributive property.

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

Step 2.12.2

Remove unnecessary parentheses.

$2 x x' - 2 x - 2 y x' + 2 y$

Step 2.12.3

Reorder terms.

$2 x x' - 2 y x' - 2 x + 2 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

$x' + 1 + 0$

Step 3.5

Add $x' + 1$ and $0$ .

$x' + 1$

Step 4

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

$2 x x' - 2 y x' - 2 x + 2 y = x' + 1$

Step 5

Solve for $x'$ .

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Step 5.1

Subtract $x'$ from both sides of the equation.

$2 x x' - 2 y x' - 2 x + 2 y - x' = 1$

Step 5.2

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

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Step 5.2.1

Add $2 x$ to both sides of the equation.

$2 x x' - 2 y x' + 2 y - x' = 1 + 2 x$

Step 5.2.2

Subtract $2 y$ from both sides of the equation.

$2 x x' - 2 y x' - x' = 1 + 2 x - 2 y$

Step 5.3

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

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Step 5.3.1

Factor $x'$ out of $2 x x'$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 5.3.5

Factor $x'$ out of $x' (2 x - 2 y) + x' \cdot - 1$ .

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

Step 5.4

Divide each term in $x' (2 x - 2 y - 1) = 1 + 2 x - 2 y$ by $2 x - 2 y - 1$ and simplify.

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Step 5.4.1

Divide each term in $x' (2 x - 2 y - 1) = 1 + 2 x - 2 y$ by $2 x - 2 y - 1$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2 x - 2 y - 1$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $x'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

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

Step 5.4.3.2

Combine the numerators over the common denominator.

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

Step 5.4.3.3

Combine the numerators over the common denominator.

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

Step 6

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

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

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