Calculus Examples

$5 x^{2} - 2 x y + 7 y^{2} = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 x^{2} - 2 x y + 7 y^{2}) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{2} - 2 x y + 7 y^{2}$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 2 x y] + \frac{d}{d x} [7 y^{2}]$ .

$\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 2 x y] + \frac{d}{d x} [7 y^{2}]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x y] + \frac{d}{d x} [7 y^{2}]$

Step 2.2.2

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

$5 (2 x) + \frac{d}{d x} [- 2 x y] + \frac{d}{d x} [7 y^{2}]$

Step 2.2.3

Multiply $2$ by $5$ .

$10 x + \frac{d}{d x} [- 2 x y] + \frac{d}{d x} [7 y^{2}]$

Step 2.3

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

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Step 2.3.1

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

$10 x - 2 \frac{d}{d x} [x y] + \frac{d}{d x} [7 y^{2}]$

Step 2.3.2

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 $g (x) = y$ .

$10 x - 2 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [7 y^{2}]$

Step 2.3.3

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

$10 x - 2 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [7 y^{2}]$

Step 2.3.4

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

$10 x - 2 (x y' + y \cdot 1) + \frac{d}{d x} [7 y^{2}]$

Step 2.3.5

Multiply $y$ by $1$ .

$10 x - 2 (x y' + y) + \frac{d}{d x} [7 y^{2}]$

Step 2.4

Evaluate $\frac{d}{d x} [7 y^{2}]$ .

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Step 2.4.1

Since $7$ is constant with respect to $x$ , the derivative of $7 y^{2}$ with respect to $x$ is $7 \frac{d}{d x} [y^{2}]$ .

$10 x - 2 (x y' + y) + 7 \frac{d}{d x} [y^{2}]$

Step 2.4.2

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 2.4.2.1

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

$10 x - 2 (x y' + y) + 7 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 2.4.2.2

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

$10 x - 2 (x y' + y) + 7 (2 u \frac{d}{d x} [y])$

Step 2.4.2.3

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

$10 x - 2 (x y' + y) + 7 (2 y \frac{d}{d x} [y])$

Step 2.4.3

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

$10 x - 2 (x y' + y) + 7 (2 y y')$

Step 2.4.4

Multiply $2$ by $7$ .

$10 x - 2 (x y' + y) + 14 y y'$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$10 x - 2 (x y') - 2 y + 14 y y'$

Step 2.5.2

Remove unnecessary parentheses.

$10 x - 2 x y' - 2 y + 14 y y'$

Step 2.5.3

Reorder terms.

$- 2 x y' + 14 y y' + 10 x - 2 y$

Step 3

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

$0$

Step 4

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

$- 2 x y' + 14 y y' + 10 x - 2 y = 0$

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 $10 x$ from both sides of the equation.

$- 2 x y' + 14 y y' - 2 y = - 10 x$

Step 5.1.2

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

$- 2 x y' + 14 y y' = - 10 x + 2 y$

Step 5.2

Factor $2 y'$ out of $- 2 x y' + 14 y y'$ .

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Step 5.2.1

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

$2 y' (- x) + 14 y y' = - 10 x + 2 y$

Step 5.2.2

Factor $2 y'$ out of $14 y y'$ .

$2 y' (- x) + 2 y' (7 y) = - 10 x + 2 y$

Step 5.2.3

Factor $2 y'$ out of $2 y' (- x) + 2 y' (7 y)$ .

$2 y' (- x + 7 y) = - 10 x + 2 y$

Step 5.3

Divide each term in $2 y' (- x + 7 y) = - 10 x + 2 y$ by $2 (- x + 7 y)$ and simplify.

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Step 5.3.1

Divide each term in $2 y' (- x + 7 y) = - 10 x + 2 y$ by $2 (- x + 7 y)$ .

$\frac{2 y' (- x + 7 y)}{2 (- x + 7 y)} = \frac{- 10 x}{2 (- x + 7 y)} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 y' (- x + 7 y)}{2 (- x + 7 y)} = \frac{- 10 x}{2 (- x + 7 y)} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (- x + 7 y)}{- x + 7 y} = \frac{- 10 x}{2 (- x + 7 y)} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.2.2

Cancel the common factor of $- x + 7 y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (- x + 7 y)}{- x + 7 y} = \frac{- 10 x}{2 (- x + 7 y)} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 10 x}{2 (- x + 7 y)} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $- 10$ and $2$ .

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Step 5.3.3.1.1.1

Factor $2$ out of $- 10 x$ .

$y' = \frac{2 (- 5 x)}{2 (- x + 7 y)} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

$y' = \frac{2 (- 5 x)}{2 (- x + 7 y)} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- 5 x}{- x + 7 y} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{5 x}{- x + 7 y} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.3.1.3

Cancel the common factor of $2$ .

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Step 5.3.3.1.3.1

Cancel the common factor.

$y' = - \frac{5 x}{- x + 7 y} + \frac{2 y}{2 (- x + 7 y)}$

Step 5.3.3.1.3.2

Rewrite the expression.

$y' = - \frac{5 x}{- x + 7 y} + \frac{y}{- x + 7 y}$

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 5 x + y}{- x + 7 y}$

Step 5.3.3.2.2

Factor $- 1$ out of $- 5 x$ .

$y' = \frac{- (5 x) + y}{- x + 7 y}$

Step 5.3.3.2.3

Factor $- 1$ out of $y$ .

$y' = \frac{- (5 x) - 1 (- y)}{- x + 7 y}$

Step 5.3.3.2.4

Factor $- 1$ out of $- (5 x) - 1 (- y)$ .

$y' = \frac{- (5 x - y)}{- x + 7 y}$

Step 5.3.3.2.5

Rewrite $- (5 x - y)$ as $- 1 (5 x - y)$ .

$y' = \frac{- 1 (5 x - y)}{- x + 7 y}$

Step 5.3.3.2.6

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

$y' = \frac{- 1 (5 x - y)}{- (x) + 7 y}$

Step 5.3.3.2.7

Factor $- 1$ out of $7 y$ .

$y' = \frac{- 1 (5 x - y)}{- (x) - (- 7 y)}$

Step 5.3.3.2.8

Factor $- 1$ out of $- (x) - (- 7 y)$ .

$y' = \frac{- 1 (5 x - y)}{- (x - 7 y)}$

Step 5.3.3.2.9

Rewrite $- (x - 7 y)$ as $- 1 (x - 7 y)$ .

$y' = \frac{- 1 (5 x - y)}{- 1 (x - 7 y)}$

Step 5.3.3.2.10

Cancel the common factor.

$y' = \frac{- 1 (5 x - y)}{- 1 (x - 7 y)}$

Step 5.3.3.2.11

Rewrite the expression.

$y' = \frac{5 x - y}{x - 7 y}$

Step 6

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

$\frac{d y}{d x} = \frac{5 x - y}{x - 7 y}$

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.

$5 x - y = 0$

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Add $y$ to both sides of the equation.

$5 x = y$

Step 7.2.2

Divide each term in $5 x = y$ by $5$ and simplify.

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Step 7.2.2.1

Divide each term in $5 x = y$ by $5$ .

$\frac{5 x}{5} = \frac{y}{5}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $5$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{y}{5}$

Step 7.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{y}{5}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify $5 {(\frac{y}{5})}^{2} - 2 \frac{y}{5} \cdot y + 7 y^{2}$ .

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Step 8.1.1

Simplify each term.

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Step 8.1.1.1

Apply the product rule to $\frac{y}{5}$ .

$5 \frac{y^{2}}{5^{2}} - 2 \frac{y}{5} \cdot y + 7 y^{2} = 0$

Step 8.1.1.2

Raise $5$ to the power of $2$ .

$5 \frac{y^{2}}{25} - 2 \frac{y}{5} \cdot y + 7 y^{2} = 0$

Step 8.1.1.3

Cancel the common factor of $5$ .

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Step 8.1.1.3.1

Factor $5$ out of $25$ .

$5 \frac{y^{2}}{5 (5)} - 2 \frac{y}{5} \cdot y + 7 y^{2} = 0$

Step 8.1.1.3.2

Cancel the common factor.

$5 \frac{y^{2}}{5 \cdot 5} - 2 \frac{y}{5} \cdot y + 7 y^{2} = 0$

Step 8.1.1.3.3

Rewrite the expression.

$\frac{y^{2}}{5} - 2 \frac{y}{5} \cdot y + 7 y^{2} = 0$

Step 8.1.1.4

Combine $- 2$ and $\frac{y}{5}$ .

$\frac{y^{2}}{5} + \frac{- 2 y}{5} \cdot y + 7 y^{2} = 0$

Step 8.1.1.5

Move the negative in front of the fraction.

$\frac{y^{2}}{5} - \frac{2 y}{5} \cdot y + 7 y^{2} = 0$

Step 8.1.1.6

Multiply $- \frac{2 y}{5} y$ .

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Step 8.1.1.6.1

Combine $y$ and $\frac{2 y}{5}$ .

$\frac{y^{2}}{5} - \frac{y (2 y)}{5} + 7 y^{2} = 0$

Step 8.1.1.6.2

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

$\frac{y^{2}}{5} - \frac{2 (y^{1} y)}{5} + 7 y^{2} = 0$

Step 8.1.1.6.3

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

$\frac{y^{2}}{5} - \frac{2 (y^{1} y^{1})}{5} + 7 y^{2} = 0$

Step 8.1.1.6.4

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

$\frac{y^{2}}{5} - \frac{2 y^{1 + 1}}{5} + 7 y^{2} = 0$

Step 8.1.1.6.5

Add $1$ and $1$ .

$\frac{y^{2}}{5} - \frac{2 y^{2}}{5} + 7 y^{2} = 0$

Step 8.1.2

Simplify terms.

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Step 8.1.2.1

Combine the numerators over the common denominator.

$7 y^{2} + \frac{y^{2} - 2 y^{2}}{5} = 0$

Step 8.1.2.2

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

$7 y^{2} + \frac{- y^{2}}{5} = 0$

Step 8.1.2.3

Move the negative in front of the fraction.

$7 y^{2} - \frac{y^{2}}{5} = 0$

Step 8.1.3

To write $7 y^{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$7 y^{2} \cdot \frac{5}{5} - \frac{y^{2}}{5} = 0$

Step 8.1.4

Simplify terms.

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Step 8.1.4.1

Combine $7 y^{2}$ and $\frac{5}{5}$ .

$\frac{7 y^{2} \cdot 5}{5} - \frac{y^{2}}{5} = 0$

Step 8.1.4.2

Combine the numerators over the common denominator.

$\frac{7 y^{2} \cdot 5 - y^{2}}{5} = 0$

Step 8.1.5

Simplify the numerator.

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Step 8.1.5.1

Factor $y^{2}$ out of $7 y^{2} \cdot 5 - y^{2}$ .

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Step 8.1.5.1.1

Factor $y^{2}$ out of $7 y^{2} \cdot 5$ .

$\frac{y^{2} (7 \cdot 5) - y^{2}}{5} = 0$

Step 8.1.5.1.2

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

$\frac{y^{2} (7 \cdot 5) + y^{2} \cdot - 1}{5} = 0$

Step 8.1.5.1.3

Factor $y^{2}$ out of $y^{2} (7 \cdot 5) + y^{2} \cdot - 1$ .

$\frac{y^{2} (7 \cdot 5 - 1)}{5} = 0$

Step 8.1.5.2

Multiply $7$ by $5$ .

$\frac{y^{2} (35 - 1)}{5} = 0$

Step 8.1.5.3

Subtract $1$ from $35$ .

$\frac{y^{2} \cdot 34}{5} = 0$

Step 8.1.6

Move $34$ to the left of $y^{2}$ .

$\frac{34 y^{2}}{5} = 0$

Step 8.2

Set the numerator equal to zero.

$34 y^{2} = 0$

Step 8.3

Solve the equation for $y$ .

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Step 8.3.1

Divide each term in $34 y^{2} = 0$ by $34$ and simplify.

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Step 8.3.1.1

Divide each term in $34 y^{2} = 0$ by $34$ .

$\frac{34 y^{2}}{34} = \frac{0}{34}$

Step 8.3.1.2

Simplify the left side.

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Step 8.3.1.2.1

Cancel the common factor of $34$ .

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Step 8.3.1.2.1.1

Cancel the common factor.

$\frac{34 y^{2}}{34} = \frac{0}{34}$

Step 8.3.1.2.1.2

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

$y^{2} = \frac{0}{34}$

Step 8.3.1.3

Simplify the right side.

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Step 8.3.1.3.1

Divide $0$ by $34$ .

$y^{2} = 0$

Step 8.3.2

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

$y = \pm \sqrt{0}$

Step 8.3.3

Simplify $\pm \sqrt{0}$ .

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Step 8.3.3.1

Rewrite $0$ as $0^{2}$ .

$y = \pm \sqrt{0^{2}}$

Step 8.3.3.2

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

$y = \pm 0$

Step 8.3.3.3

Plus or minus $0$ is $0$ .

$y = 0$

Step 9

Solve for $x$ when $y$ is $0$ .

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Step 9.1

Remove parentheses.

$x = \frac{0}{5}$

Step 9.2

Divide $0$ by $5$ .

$x = 0$

Step 10

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

$(0, 0)$

Step 11