Calculus Examples

$x^{2} + x y + y^{2} = 27$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + x y + y^{2}) = \frac{d}{d x} (27)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $y$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.2

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

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

Step 2.4

Reorder terms.

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

Step 3

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$0$

Step 4

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

$x y' + 2 y y' + 2 x + y = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Subtract $y$ from both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x + 2 y$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

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

Step 5.3.3.2

Factor $- 1$ out of $- 2 x$ .

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

Step 5.3.3.3

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

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

Step 5.3.3.4

Factor $- 1$ out of $- (2 x) - (y)$ .

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

Step 5.3.3.5

Simplify the expression.

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Step 5.3.3.5.1

Rewrite $- (2 x + y)$ as $- 1 (2 x + y)$ .

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

Step 5.3.3.5.2

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$2 x + y = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Subtract $y$ from both sides of the equation.

$2 x = - y$

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative in front of the fraction.

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

Step 8

Solve for $y$ .

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Step 8.1

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

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.1.1.1.1

Apply the product rule to $- \frac{y}{2}$ .

${(- 1)}^{2} {(\frac{y}{2})}^{2} + (- \frac{y}{2}) y + y^{2} = 27$

Step 8.1.1.1.2

Apply the product rule to $\frac{y}{2}$ .

${(- 1)}^{2} \frac{y^{2}}{2^{2}} + (- \frac{y}{2}) y + y^{2} = 27$

Step 8.1.1.2

Raise $- 1$ to the power of $2$ .

$1 \frac{y^{2}}{2^{2}} + (- \frac{y}{2}) y + y^{2} = 27$

Step 8.1.1.3

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

$\frac{y^{2}}{2^{2}} + (- \frac{y}{2}) y + y^{2} = 27$

Step 8.1.1.4

Raise $2$ to the power of $2$ .

$\frac{y^{2}}{4} + (- \frac{y}{2}) y + y^{2} = 27$

Step 8.1.1.5

Multiply $(- \frac{y}{2}) y$ .

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Step 8.1.1.5.1

Combine $y$ and $\frac{y}{2}$ .

$\frac{y^{2}}{4} - \frac{y \cdot y}{2} + y^{2} = 27$

Step 8.1.1.5.2

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

$\frac{y^{2}}{4} - \frac{y^{1} y}{2} + y^{2} = 27$

Step 8.1.1.5.3

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

$\frac{y^{2}}{4} - \frac{y^{1} y^{1}}{2} + y^{2} = 27$

Step 8.1.1.5.4

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

$\frac{y^{2}}{4} - \frac{y^{1 + 1}}{2} + y^{2} = 27$

Step 8.1.1.5.5

Add $1$ and $1$ .

$\frac{y^{2}}{4} - \frac{y^{2}}{2} + y^{2} = 27$

Step 8.1.2

Find the common denominator.

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Step 8.1.2.1

Multiply $\frac{y^{2}}{2}$ by $\frac{2}{2}$ .

$\frac{y^{2}}{4} - (\frac{y^{2}}{2} \cdot \frac{2}{2}) + y^{2} = 27$

Step 8.1.2.2

Multiply $\frac{y^{2}}{2}$ by $\frac{2}{2}$ .

$\frac{y^{2}}{4} - \frac{y^{2} \cdot 2}{2 \cdot 2} + y^{2} = 27$

Step 8.1.2.3

Write $y^{2}$ as a fraction with denominator $1$ .

$\frac{y^{2}}{4} - \frac{y^{2} \cdot 2}{2 \cdot 2} + \frac{y^{2}}{1} = 27$

Step 8.1.2.4

Multiply $\frac{y^{2}}{1}$ by $\frac{4}{4}$ .

$\frac{y^{2}}{4} - \frac{y^{2} \cdot 2}{2 \cdot 2} + \frac{y^{2}}{1} \cdot \frac{4}{4} = 27$

Step 8.1.2.5

Multiply $\frac{y^{2}}{1}$ by $\frac{4}{4}$ .

$\frac{y^{2}}{4} - \frac{y^{2} \cdot 2}{2 \cdot 2} + \frac{y^{2} \cdot 4}{4} = 27$

Step 8.1.2.6

Multiply $2$ by $2$ .

$\frac{y^{2}}{4} - \frac{y^{2} \cdot 2}{4} + \frac{y^{2} \cdot 4}{4} = 27$

Step 8.1.3

Combine the numerators over the common denominator.

$\frac{y^{2} - y^{2} \cdot 2 + y^{2} \cdot 4}{4} = 27$

Step 8.1.4

Simplify each term.

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Step 8.1.4.1

Multiply $2$ by $- 1$ .

$\frac{y^{2} - 2 y^{2} + y^{2} \cdot 4}{4} = 27$

Step 8.1.4.2

Move $4$ to the left of $y^{2}$ .

$\frac{y^{2} - 2 y^{2} + 4 y^{2}}{4} = 27$

Step 8.1.5

Simplify by adding terms.

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Step 8.1.5.1

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

$\frac{- y^{2} + 4 y^{2}}{4} = 27$

Step 8.1.5.2

Add $- y^{2}$ and $4 y^{2}$ .

$\frac{3 y^{2}}{4} = 27$

Step 8.2

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 y^{2}}{4} = \frac{4}{3} \cdot 27$

Step 8.3

Simplify both sides of the equation.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Simplify $\frac{4}{3} \cdot \frac{3 y^{2}}{4}$ .

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Step 8.3.1.1.1

Combine.

$\frac{4 (3 y^{2})}{3 \cdot 4} = \frac{4}{3} \cdot 27$

Step 8.3.1.1.2

Cancel the common factor of $4$ .

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Step 8.3.1.1.2.1

Cancel the common factor.

$\frac{4 (3 y^{2})}{3 \cdot 4} = \frac{4}{3} \cdot 27$

Step 8.3.1.1.2.2

Rewrite the expression.

$\frac{3 y^{2}}{3} = \frac{4}{3} \cdot 27$

Step 8.3.1.1.3

Cancel the common factor of $3$ .

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Step 8.3.1.1.3.1

Cancel the common factor.

$\frac{3 y^{2}}{3} = \frac{4}{3} \cdot 27$

Step 8.3.1.1.3.2

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

$y^{2} = \frac{4}{3} \cdot 27$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Simplify $\frac{4}{3} \cdot 27$ .

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Step 8.3.2.1.1

Cancel the common factor of $3$ .

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Step 8.3.2.1.1.1

Factor $3$ out of $27$ .

$y^{2} = \frac{4}{3} \cdot (3 (9))$

Step 8.3.2.1.1.2

Cancel the common factor.

$y^{2} = \frac{4}{3} \cdot (3 \cdot 9)$

Step 8.3.2.1.1.3

Rewrite the expression.

$y^{2} = 4 \cdot 9$

Step 8.3.2.1.2

Multiply $4$ by $9$ .

$y^{2} = 36$

Step 8.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{36}$

Step 8.5

Simplify $\pm \sqrt{36}$ .

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Step 8.5.1

Rewrite $36$ as $6^{2}$ .

$y = \pm \sqrt{6^{2}}$

Step 8.5.2

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 6$

Step 8.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 8.6.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 6$

Step 8.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 6$

Step 8.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 6, - 6$

Step 9

Solve for $x$ when $y$ is $6$ .

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Step 9.1

Remove parentheses.

$x = - \frac{6}{2}$

Step 9.2

Simplify $- \frac{6}{2}$ .

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Step 9.2.1

Divide $6$ by $2$ .

$x = - 1 \cdot 3$

Step 9.2.2

Multiply $- 1$ by $3$ .

$x = - 3$

Step 10

Simplify $- \frac{- 6}{2}$ .

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Step 10.1

Divide $- 6$ by $2$ .

$x = - - 3$

Step 10.2

Multiply $- 1$ by $- 3$ .

$x = 3$

Step 11

Find the points where $\frac{d y}{d x} = 0$ .

$(- 3, 6)$

$(3, - 6)$

Step 12