Calculus Examples

$x^{2} + 2 y^{2} + 2 x y + 2 x + 4 y + 7$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{2} + 2 y^{2} + 2 x y + 2 x + 4 y + 7$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [2 x y] + \frac{d}{d x} [2 x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [2 x y] + \frac{d}{d x} [2 x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.1.2

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

$2 x + \frac{d}{d x} [2 y^{2}] + \frac{d}{d x} [2 x y] + \frac{d}{d x} [2 x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.1.3

Since $2 y^{2}$ is constant with respect to $x$ , the derivative of $2 y^{2}$ with respect to $x$ is $0$ .

$2 x + 0 + \frac{d}{d x} [2 x y] + \frac{d}{d x} [2 x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.2

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

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Step 1.1.2.1

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

$2 x + 0 + 2 y \frac{d}{d x} [x] + \frac{d}{d x} [2 x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.2.2

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

$2 x + 0 + 2 y \cdot 1 + \frac{d}{d x} [2 x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.2.3

Multiply $2$ by $1$ .

$2 x + 0 + 2 y + \frac{d}{d x} [2 x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.3

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

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Step 1.1.3.1

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

$2 x + 0 + 2 y + 2 \frac{d}{d x} [x] + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.3.2

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

$2 x + 0 + 2 y + 2 \cdot 1 + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 x + 0 + 2 y + 2 + \frac{d}{d x} [4 y] + \frac{d}{d x} [7]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $4 y$ is constant with respect to $x$ , the derivative of $4 y$ with respect to $x$ is $0$ .

$2 x + 0 + 2 y + 2 + 0 + \frac{d}{d x} [7]$

Step 1.1.4.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$2 x + 0 + 2 y + 2 + 0 + 0$

Step 1.1.5

Combine terms.

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Step 1.1.5.1

Add $2 x$ and $0$ .

$2 x + 2 y + 2 + 0 + 0$

Step 1.1.5.2

Add $2 x + 2 y + 2$ and $0$ .

$2 x + 2 y + 2 + 0$

Step 1.1.5.3

Add $2 x + 2 y + 2$ and $0$ .

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

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x + 2 y + 2$ .

$2 x + 2 y + 2$

Step 2

Set the first derivative equal to $0$ then solve the equation $2 x + 2 y + 2 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$2 x + 2 y + 2 = 0$

Step 2.2

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

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Step 2.2.1

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

$2 x + 2 = - 2 y$

Step 2.2.2

Subtract $2$ from both sides of the equation.

$2 x = - 2 y - 2$

Step 2.3

Divide each term in $2 x = - 2 y - 2$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 x = - 2 y - 2$ by $2$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Cancel the common factor of $- 2$ and $2$ .

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Step 2.3.3.1.1.1

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

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

Step 2.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.1.1.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.1.1.2.2

Cancel the common factor.

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

Step 2.3.3.1.1.2.3

Rewrite the expression.

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

Step 2.3.3.1.1.2.4

Divide $- y$ by $1$ .

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

Step 2.3.3.1.2

Divide $- 2$ by $2$ .

$x = - y - 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $x^{2} + 2 y^{2} + 2 x y + 2 x + 4 y + 7$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - y - 1$ .

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Step 4.1.1

Substitute $- y - 1$ for $x$ .

${(- y - 1)}^{2} + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

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

$(- y - 1) (- y - 1) + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.2

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

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Step 4.1.2.1.2.1

Apply the distributive property.

$- y (- y - 1) - 1 (- y - 1) + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.2.2

Apply the distributive property.

$- y (- y) - y \cdot - 1 - 1 (- y - 1) + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.2.3

Apply the distributive property.

$- y (- y) - y \cdot - 1 - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3

Simplify and combine like terms.

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Step 4.1.2.1.3.1

Simplify each term.

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Step 4.1.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot - 1 y \cdot y - y \cdot - 1 - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.2

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

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Step 4.1.2.1.3.1.2.1

Move $y$ .

$- 1 \cdot - 1 (y \cdot y) - y \cdot - 1 - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.2.2

Multiply $y$ by $y$ .

$- 1 \cdot - 1 y^{2} - y \cdot - 1 - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.3

Multiply $- 1$ by $- 1$ .

$1 y^{2} - y \cdot - 1 - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.4

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

$y^{2} - y \cdot - 1 - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.5

Multiply $- y \cdot - 1$ .

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Step 4.1.2.1.3.1.5.1

Multiply $- 1$ by $- 1$ .

$y^{2} + 1 y - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.5.2

Multiply $y$ by $1$ .

$y^{2} + y - 1 (- y) - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.6

Multiply $- 1 (- y)$ .

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Step 4.1.2.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$y^{2} + y + 1 y - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.6.2

Multiply $y$ by $1$ .

$y^{2} + y + y - 1 \cdot - 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.1.7

Multiply $- 1$ by $- 1$ .

$y^{2} + y + y + 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.3.2

Add $y$ and $y$ .

$y^{2} + 2 y + 1 + 2 y^{2} + 2 (- y - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.4

Apply the distributive property.

$y^{2} + 2 y + 1 + 2 y^{2} + (2 (- y) + 2 \cdot - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.5

Multiply $- 1$ by $2$ .

$y^{2} + 2 y + 1 + 2 y^{2} + (- 2 y + 2 \cdot - 1) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.6

Multiply $2$ by $- 1$ .

$y^{2} + 2 y + 1 + 2 y^{2} + (- 2 y - 2) \cdot y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.7

Apply the distributive property.

$y^{2} + 2 y + 1 + 2 y^{2} - 2 y \cdot y - 2 y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.8

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

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Step 4.1.2.1.8.1

Move $y$ .

$y^{2} + 2 y + 1 + 2 y^{2} - 2 (y \cdot y) - 2 y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.8.2

Multiply $y$ by $y$ .

$y^{2} + 2 y + 1 + 2 y^{2} - 2 y^{2} - 2 y + 2 (- y - 1) + 4 y + 7$

Step 4.1.2.1.9

Apply the distributive property.

$y^{2} + 2 y + 1 + 2 y^{2} - 2 y^{2} - 2 y + 2 (- y) + 2 \cdot - 1 + 4 y + 7$

Step 4.1.2.1.10

Multiply $- 1$ by $2$ .

$y^{2} + 2 y + 1 + 2 y^{2} - 2 y^{2} - 2 y - 2 y + 2 \cdot - 1 + 4 y + 7$

Step 4.1.2.1.11

Multiply $2$ by $- 1$ .

$y^{2} + 2 y + 1 + 2 y^{2} - 2 y^{2} - 2 y - 2 y - 2 + 4 y + 7$

Step 4.1.2.2

Simplify by adding terms.

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Step 4.1.2.2.1

Combine the opposite terms in $y^{2} + 2 y + 1 + 2 y^{2} - 2 y^{2} - 2 y - 2 y - 2 + 4 y + 7$ .

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Step 4.1.2.2.1.1

Subtract $2 y^{2}$ from $2 y^{2}$ .

$y^{2} + 2 y + 1 + 0 - 2 y - 2 y - 2 + 4 y + 7$

Step 4.1.2.2.1.2

Add $y^{2} + 2 y + 1$ and $0$ .

$y^{2} + 2 y + 1 - 2 y - 2 y - 2 + 4 y + 7$

Step 4.1.2.2.1.3

Subtract $2 y$ from $2 y$ .

$y^{2} + 0 + 1 - 2 y - 2 + 4 y + 7$

Step 4.1.2.2.1.4

Add $y^{2}$ and $0$ .

$y^{2} + 1 - 2 y - 2 + 4 y + 7$

Step 4.1.2.2.2

Subtract $2$ from $1$ .

$y^{2} - 2 y - 1 + 4 y + 7$

Step 4.1.2.2.3

Add $- 2 y$ and $4 y$ .

$y^{2} + 2 y - 1 + 7$

Step 4.1.2.2.4

Add $- 1$ and $7$ .

$y^{2} + 2 y + 6$

Step 4.2

List all of the points.

$(- y - 1, y^{2} + 2 y + 6)$

Step 5