Calculus Examples

$x^{2} + x y = 10$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + x y) = \frac{d}{d x} (10)$

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 $x^{2} + x y$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x y]$ .

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

Step 2.1.2

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

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

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 $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

$2 x + x y' + y \cdot 1$

Step 2.2.4

Multiply $y$ by $1$ .

$2 x + x y' + y$

Step 2.3

Reorder terms.

$x y' + 2 x + y$

Step 3

Since $10$ is constant with respect to $x$ , the derivative of $10$ with respect to $x$ is $0$ .

$0$

Step 4

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

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

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.

$x y' + y = - 2 x$

Step 5.1.2

Subtract $y$ from both sides of the equation.

$x y' = - 2 x - y$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $x$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $x$ .

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Step 5.2.3.1.1.1

Cancel the common factor.

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

Step 5.2.3.1.1.2

Divide $- 2$ by $1$ .

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

Step 5.2.3.1.2

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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Step 7.1

Add $2$ to both sides of the equation.

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

Step 7.2

Find the LCD of the terms in the equation.

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Step 7.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 7.2.2

The LCM of one and any expression is the expression.

$x$

Step 7.3

Multiply each term in $- \frac{y}{x} = 2$ by $x$ to eliminate the fractions.

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Step 7.3.1

Multiply each term in $- \frac{y}{x} = 2$ by $x$ .

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

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $x$ .

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Step 7.3.2.1.1

Move the leading negative in $- \frac{y}{x}$ into the numerator.

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

Step 7.3.2.1.2

Cancel the common factor.

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

Step 7.3.2.1.3

Rewrite the expression.

$- y = 2 x$

Step 7.4

Solve the equation.

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Step 7.4.1

Rewrite the equation as $2 x = - y$ .

$2 x = - y$

Step 7.4.2

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

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Step 7.4.2.1

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

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

Step 7.4.2.2

Simplify the left side.

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Step 7.4.2.2.1

Cancel the common factor of $2$ .

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Step 7.4.2.2.1.1

Cancel the common factor.

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

Step 7.4.2.2.1.2

Divide $x$ by $1$ .

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

Step 7.4.2.3

Simplify the right side.

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Step 7.4.2.3.1

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

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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 $- \frac{y}{2}$ .

${(- 1)}^{2} {(\frac{y}{2})}^{2} + (- \frac{y}{2}) y = 10$

Step 8.1.1.1.2

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

${(- 1)}^{2} \frac{y^{2}}{2^{2}} + (- \frac{y}{2}) y = 10$

Step 8.1.1.2

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

$1 \frac{y^{2}}{2^{2}} + (- \frac{y}{2}) y = 10$

Step 8.1.1.3

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

$\frac{y^{2}}{2^{2}} + (- \frac{y}{2}) y = 10$

Step 8.1.1.4

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

$\frac{y^{2}}{4} + (- \frac{y}{2}) y = 10$

Step 8.1.1.5

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

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Step 8.1.1.5.1

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

$\frac{y^{2}}{4} - \frac{y \cdot y}{2} = 10$

Step 8.1.1.5.2

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

$\frac{y^{2}}{4} - \frac{y^{1} y}{2} = 10$

Step 8.1.1.5.3

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

$\frac{y^{2}}{4} - \frac{y^{1} y^{1}}{2} = 10$

Step 8.1.1.5.4

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

$\frac{y^{2}}{4} - \frac{y^{1 + 1}}{2} = 10$

Step 8.1.1.5.5

Add $1$ and $1$ .

$\frac{y^{2}}{4} - \frac{y^{2}}{2} = 10$

Step 8.1.2

To write $- \frac{y^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{y^{2}}{4} - \frac{y^{2}}{2} \cdot \frac{2}{2} = 10$

Step 8.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1.3.1

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

$\frac{y^{2}}{4} - \frac{y^{2} \cdot 2}{2 \cdot 2} = 10$

Step 8.1.3.2

Multiply $2$ by $2$ .

$\frac{y^{2}}{4} - \frac{y^{2} \cdot 2}{4} = 10$

Step 8.1.4

Combine the numerators over the common denominator.

$\frac{y^{2} - y^{2} \cdot 2}{4} = 10$

Step 8.1.5

Simplify the numerator.

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Step 8.1.5.1

Factor $y^{2}$ out of $y^{2} - y^{2} \cdot 2$ .

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Step 8.1.5.1.1

Multiply by $1$ .

$\frac{y^{2} \cdot 1 - y^{2} \cdot 2}{4} = 10$

Step 8.1.5.1.2

Factor $y^{2}$ out of $- y^{2} \cdot 2$ .

$\frac{y^{2} \cdot 1 + y^{2} (- 1 \cdot 2)}{4} = 10$

Step 8.1.5.1.3

Factor $y^{2}$ out of $y^{2} \cdot 1 + y^{2} (- 1 \cdot 2)$ .

$\frac{y^{2} (1 - 1 \cdot 2)}{4} = 10$

Step 8.1.5.2

Multiply $- 1$ by $2$ .

$\frac{y^{2} (1 - 2)}{4} = 10$

Step 8.1.5.3

Subtract $2$ from $1$ .

$\frac{y^{2} \cdot - 1}{4} = 10$

Step 8.1.6

Simplify the expression.

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Step 8.1.6.1

Move $- 1$ to the left of $y^{2}$ .

$\frac{- 1 \cdot y^{2}}{4} = 10$

Step 8.1.6.2

Move the negative in front of the fraction.

$- \frac{y^{2}}{4} = 10$

Step 8.2

Multiply both sides of the equation by $- 4$ .

$- 4 (- \frac{y^{2}}{4}) = - 4 \cdot 10$

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 $- 4 (- \frac{y^{2}}{4})$ .

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Step 8.3.1.1.1

Cancel the common factor of $4$ .

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Step 8.3.1.1.1.1

Move the leading negative in $- \frac{y^{2}}{4}$ into the numerator.

$- 4 \frac{- y^{2}}{4} = - 4 \cdot 10$

Step 8.3.1.1.1.2

Factor $4$ out of $- 4$ .

$4 (- 1) \frac{- y^{2}}{4} = - 4 \cdot 10$

Step 8.3.1.1.1.3

Cancel the common factor.

$4 \cdot - 1 \frac{- y^{2}}{4} = - 4 \cdot 10$

Step 8.3.1.1.1.4

Rewrite the expression.

$- - y^{2} = - 4 \cdot 10$

Step 8.3.1.1.2

Multiply.

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Step 8.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 y^{2} = - 4 \cdot 10$

Step 8.3.1.1.2.2

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

$y^{2} = - 4 \cdot 10$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Multiply $- 4$ by $10$ .

$y^{2} = - 40$

Step 8.4

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

$y = \pm \sqrt{- 40}$

Step 8.5

Simplify $\pm \sqrt{- 40}$ .

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Step 8.5.1

Rewrite $- 40$ as $- 1 (40)$ .

$y = \pm \sqrt{- 1 (40)}$

Step 8.5.2

Rewrite $\sqrt{- 1 (40)}$ as $\sqrt{- 1} \cdot \sqrt{40}$ .

$y = \pm \sqrt{- 1} \cdot \sqrt{40}$

Step 8.5.3

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \pm i \cdot \sqrt{40}$

Step 8.5.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 8.5.4.1

Factor $4$ out of $40$ .

$y = \pm i \cdot \sqrt{4 (10)}$

Step 8.5.4.2

Rewrite $4$ as $2^{2}$ .

$y = \pm i \cdot \sqrt{2^{2} \cdot 10}$

Step 8.5.5

Pull terms out from under the radical.

$y = \pm i \cdot (2 \sqrt{10})$

Step 8.5.6

Move $2$ to the left of $i$ .

$y = \pm 2 i \sqrt{10}$

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.

$y = 2 i \sqrt{10}$

Step 8.6.2

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

$y = - 2 i \sqrt{10}$

Step 8.6.3

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

$y = 2 i \sqrt{10}, - 2 i \sqrt{10}$

Step 9

Solve for $x$ when $y$ is $2 i \sqrt{10}$ .

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Step 9.1

Remove parentheses.

$x = - \frac{2 i \sqrt{10}}{2}$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Cancel the common factor.

$x = - \frac{2 i \sqrt{10}}{2}$

Step 9.2.2

Divide $i \sqrt{10}$ by $1$ .

$x = - (i \sqrt{10})$

$x = - i \sqrt{10}$

Step 10

Simplify $- \frac{- 2 i \sqrt{10}}{2}$ .

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Step 10.1

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

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Step 10.1.1

Factor $2$ out of $- 2 i \sqrt{10}$ .

$x = - \frac{2 (- i \sqrt{10})}{2}$

Step 10.1.2

Cancel the common factors.

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Step 10.1.2.1

Factor $2$ out of $2$ .

$x = - \frac{2 (- i \sqrt{10})}{2 (1)}$

Step 10.1.2.2

Cancel the common factor.

$x = - \frac{2 (- i \sqrt{10})}{2 \cdot 1}$

Step 10.1.2.3

Rewrite the expression.

$x = - \frac{- i \sqrt{10}}{1}$

Step 10.1.2.4

Divide $- i \sqrt{10}$ by $1$ .

$x = - (- i \sqrt{10})$

Step 10.2

Multiply $- (- i \sqrt{10})$ .

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Step 10.2.1

Multiply $- 1$ by $- 1$ .

$x = 1 (i \sqrt{10})$

Step 10.2.2

Multiply $\sqrt{10}$ by $1$ .

$x = \sqrt{10} i$

Step 11

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

$(- i \sqrt{10}, 2 i \sqrt{10})$

$(\sqrt{10} i, - 2 i \sqrt{10})$

Step 12