Calculus Examples

$x^{3} + y^{3} - 6 x y = 0$ , $(\frac{4}{3}, \frac{8}{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (x^{3} + y^{3} - 6 x y) = \frac{d}{d y} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + y^{3} - 6 x y$ with respect to $y$ is $\frac{d}{d y} [x^{3}] + \frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 x y]$ .

$\frac{d}{d y} [x^{3}] + \frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 x y]$

Step 2.2

Evaluate $\frac{d}{d y} [x^{3}]$ .

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Step 2.2.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{3}$ and $g (y) = x$ .

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Step 2.2.1.1

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

$\frac{d}{d u} [u^{3}] \frac{d}{d y} [x] + \frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 x y]$

Step 2.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d y} [x] + \frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 x y]$

Step 2.2.1.3

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

$3 x^{2} \frac{d}{d y} [x] + \frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 x y]$

Step 2.2.2

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

$3 x^{2} x' + \frac{d}{d y} [y^{3}] + \frac{d}{d y} [- 6 x y]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 3$ .

$3 x^{2} x' + 3 y^{2} + \frac{d}{d y} [- 6 x y]$

Step 2.4

Evaluate $\frac{d}{d y} [- 6 x y]$ .

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Step 2.4.1

Since $- 6$ is constant with respect to $y$ , the derivative of $- 6 x y$ with respect to $y$ is $- 6 \frac{d}{d y} [x y]$ .

$3 x^{2} x' + 3 y^{2} - 6 \frac{d}{d y} [x y]$

Step 2.4.2

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 $g (y) = y$ .

$3 x^{2} x' + 3 y^{2} - 6 (x \frac{d}{d y} [y] + y \frac{d}{d y} [x])$

Step 2.4.3

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

$3 x^{2} x' + 3 y^{2} - 6 (x \cdot 1 + y \frac{d}{d y} [x])$

Step 2.4.4

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

$3 x^{2} x' + 3 y^{2} - 6 (x \cdot 1 + y x')$

Step 2.4.5

Multiply $x$ by $1$ .

$3 x^{2} x' + 3 y^{2} - 6 (x + y x')$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$3 x^{2} x' + 3 y^{2} - 6 x - 6 (y x')$

Step 2.5.2

Remove unnecessary parentheses.

$3 x^{2} x' + 3 y^{2} - 6 x - 6 y x'$

Step 2.5.3

Reorder terms.

$3 y^{2} + 3 x^{2} x' - 6 y x' - 6 x$

Step 3

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

$0$

Step 4

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

$3 y^{2} + 3 x^{2} x' - 6 y x' - 6 x = 0$

Step 5

Solve for $x'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $3 y^{2}$ from both sides of the equation.

$3 x^{2} x' - 6 y x' - 6 x = - 3 y^{2}$

Step 5.1.2

Add $6 x$ to both sides of the equation.

$3 x^{2} x' - 6 y x' = - 3 y^{2} + 6 x$

Step 5.2

Factor $3 x'$ out of $3 x^{2} x' - 6 y x'$ .

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Step 5.2.1

Factor $3 x'$ out of $3 x^{2} x'$ .

$3 x' x^{2} - 6 y x' = - 3 y^{2} + 6 x$

Step 5.2.2

Factor $3 x'$ out of $- 6 y x'$ .

$3 x' x^{2} + 3 x' (- 2 y) = - 3 y^{2} + 6 x$

Step 5.2.3

Factor $3 x'$ out of $3 x' x^{2} + 3 x' (- 2 y)$ .

$3 x' (x^{2} - 2 y) = - 3 y^{2} + 6 x$

Step 5.3

Divide each term in $3 x' (x^{2} - 2 y) = - 3 y^{2} + 6 x$ by $3 (x^{2} - 2 y)$ and simplify.

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Step 5.3.1

Divide each term in $3 x' (x^{2} - 2 y) = - 3 y^{2} + 6 x$ by $3 (x^{2} - 2 y)$ .

$\frac{3 x' (x^{2} - 2 y)}{3 (x^{2} - 2 y)} = \frac{- 3 y^{2}}{3 (x^{2} - 2 y)} + \frac{6 x}{3 (x^{2} - 2 y)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 x' (x^{2} - 2 y)}{3 (x^{2} - 2 y)} = \frac{- 3 y^{2}}{3 (x^{2} - 2 y)} + \frac{6 x}{3 (x^{2} - 2 y)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x' (x^{2} - 2 y)}{x^{2} - 2 y} = \frac{- 3 y^{2}}{3 (x^{2} - 2 y)} + \frac{6 x}{3 (x^{2} - 2 y)}$

Step 5.3.2.2

Cancel the common factor of $x^{2} - 2 y$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{x' (x^{2} - 2 y)}{x^{2} - 2 y} = \frac{- 3 y^{2}}{3 (x^{2} - 2 y)} + \frac{6 x}{3 (x^{2} - 2 y)}$

Step 5.3.2.2.2

Divide $x'$ by $1$ .

$x' = \frac{- 3 y^{2}}{3 (x^{2} - 2 y)} + \frac{6 x}{3 (x^{2} - 2 y)}$

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 $- 3$ and $3$ .

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Step 5.3.3.1.1.1

Factor $3$ out of $- 3 y^{2}$ .

$x' = \frac{3 (- y^{2})}{3 (x^{2} - 2 y)} + \frac{6 x}{3 (x^{2} - 2 y)}$

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.

$x' = \frac{3 (- y^{2})}{3 (x^{2} - 2 y)} + \frac{6 x}{3 (x^{2} - 2 y)}$

Step 5.3.3.1.1.2.2

Rewrite the expression.

$x' = \frac{- y^{2}}{x^{2} - 2 y} + \frac{6 x}{3 (x^{2} - 2 y)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$x' = - \frac{y^{2}}{x^{2} - 2 y} + \frac{6 x}{3 (x^{2} - 2 y)}$

Step 5.3.3.1.3

Cancel the common factor of $6$ and $3$ .

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Step 5.3.3.1.3.1

Factor $3$ out of $6 x$ .

$x' = - \frac{y^{2}}{x^{2} - 2 y} + \frac{3 (2 x)}{3 (x^{2} - 2 y)}$

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

$x' = - \frac{y^{2}}{x^{2} - 2 y} + \frac{3 (2 x)}{3 (x^{2} - 2 y)}$

Step 5.3.3.1.3.2.2

Rewrite the expression.

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

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

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

Step 5.3.3.2.2

Factor $- 1$ out of $- y^{2}$ .

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

Step 5.3.3.2.3

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

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

Step 5.3.3.2.4

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

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

Step 5.3.3.2.5

Simplify the expression.

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Step 5.3.3.2.5.1

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

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

Step 5.3.3.2.5.2

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Replace $x$ with $\frac{4}{3}$ and $y$ with $\frac{8}{3}$ in the expression.

$\frac{d x}{d y} = - \frac{{(\frac{8}{3})}^{2} - 2 (\frac{4}{3})}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8

Simplify the result.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Apply the product rule to $\frac{8}{3}$ .

$- \frac{\frac{8^{2}}{3^{2}} - 2 (\frac{4}{3})}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.2

Raise $8$ to the power of $2$ .

$- \frac{\frac{64}{3^{2}} - 2 (\frac{4}{3})}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.3

Raise $3$ to the power of $2$ .

$- \frac{\frac{64}{9} - 2 (\frac{4}{3})}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.4

Multiply $- 2 (\frac{4}{3})$ .

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Step 8.1.4.1

Combine $- 2$ and $\frac{4}{3}$ .

$- \frac{\frac{64}{9} + \frac{- 2 \cdot 4}{3}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.4.2

Multiply $- 2$ by $4$ .

$- \frac{\frac{64}{9} + \frac{- 8}{3}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.5

Move the negative in front of the fraction.

$- \frac{\frac{64}{9} - \frac{8}{3}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.6

To write $- \frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{\frac{64}{9} - \frac{8}{3} \cdot \frac{3}{3}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.7

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1.7.1

Multiply $\frac{8}{3}$ by $\frac{3}{3}$ .

$- \frac{\frac{64}{9} - \frac{8 \cdot 3}{3 \cdot 3}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.7.2

Multiply $3$ by $3$ .

$- \frac{\frac{64}{9} - \frac{8 \cdot 3}{9}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.8

Combine the numerators over the common denominator.

$- \frac{\frac{64 - 8 \cdot 3}{9}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.9

Simplify the numerator.

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Step 8.1.9.1

Multiply $- 8$ by $3$ .

$- \frac{\frac{64 - 24}{9}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.1.9.2

Subtract $24$ from $64$ .

$- \frac{\frac{40}{9}}{{(\frac{4}{3})}^{2} - 2 (\frac{8}{3})}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Apply the product rule to $\frac{4}{3}$ .

$- \frac{\frac{40}{9}}{\frac{4^{2}}{3^{2}} - 2 (\frac{8}{3})}$

Step 8.2.2

Raise $4$ to the power of $2$ .

$- \frac{\frac{40}{9}}{\frac{16}{3^{2}} - 2 (\frac{8}{3})}$

Step 8.2.3

Raise $3$ to the power of $2$ .

$- \frac{\frac{40}{9}}{\frac{16}{9} - 2 (\frac{8}{3})}$

Step 8.2.4

Multiply $- 2 (\frac{8}{3})$ .

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Step 8.2.4.1

Combine $- 2$ and $\frac{8}{3}$ .

$- \frac{\frac{40}{9}}{\frac{16}{9} + \frac{- 2 \cdot 8}{3}}$

Step 8.2.4.2

Multiply $- 2$ by $8$ .

$- \frac{\frac{40}{9}}{\frac{16}{9} + \frac{- 16}{3}}$

Step 8.2.5

Move the negative in front of the fraction.

$- \frac{\frac{40}{9}}{\frac{16}{9} - \frac{16}{3}}$

Step 8.2.6

To write $- \frac{16}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{\frac{40}{9}}{\frac{16}{9} - \frac{16}{3} \cdot \frac{3}{3}}$

Step 8.2.7

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.2.7.1

Multiply $\frac{16}{3}$ by $\frac{3}{3}$ .

$- \frac{\frac{40}{9}}{\frac{16}{9} - \frac{16 \cdot 3}{3 \cdot 3}}$

Step 8.2.7.2

Multiply $3$ by $3$ .

$- \frac{\frac{40}{9}}{\frac{16}{9} - \frac{16 \cdot 3}{9}}$

Step 8.2.8

Combine the numerators over the common denominator.

$- \frac{\frac{40}{9}}{\frac{16 - 16 \cdot 3}{9}}$

Step 8.2.9

Simplify the numerator.

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Step 8.2.9.1

Multiply $- 16$ by $3$ .

$- \frac{\frac{40}{9}}{\frac{16 - 48}{9}}$

Step 8.2.9.2

Subtract $48$ from $16$ .

$- \frac{\frac{40}{9}}{\frac{- 32}{9}}$

Step 8.2.10

Move the negative in front of the fraction.

$- \frac{\frac{40}{9}}{- \frac{32}{9}}$

Step 8.3

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{40}{9} (- \frac{9}{32}))$

Step 8.4

Cancel the common factor of $8$ .

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Step 8.4.1

Move the leading negative in $- \frac{9}{32}$ into the numerator.

$- (\frac{40}{9} \cdot \frac{- 9}{32})$

Step 8.4.2

Factor $8$ out of $40$ .

$- (\frac{8 (5)}{9} \cdot \frac{- 9}{32})$

Step 8.4.3

Factor $8$ out of $32$ .

$- (\frac{8 \cdot 5}{9} \cdot \frac{- 9}{8 \cdot 4})$

Step 8.4.4

Cancel the common factor.

$- (\frac{8 \cdot 5}{9} \cdot \frac{- 9}{8 \cdot 4})$

Step 8.4.5

Rewrite the expression.

$- (\frac{5}{9} \cdot \frac{- 9}{4})$

Step 8.5

Cancel the common factor of $9$ .

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Step 8.5.1

Factor $9$ out of $- 9$ .

$- (\frac{5}{9} \cdot \frac{9 (- 1)}{4})$

Step 8.5.2

Cancel the common factor.

$- (\frac{5}{9} \cdot \frac{9 \cdot - 1}{4})$

Step 8.5.3

Rewrite the expression.

$- (5 (\frac{- 1}{4}))$

Step 8.6

Combine $5$ and $\frac{- 1}{4}$ .

$- \frac{5 \cdot - 1}{4}$

Step 8.7

Simplify the expression.

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Step 8.7.1

Multiply $5$ by $- 1$ .

$- \frac{- 5}{4}$

Step 8.7.2

Move the negative in front of the fraction.

$- - \frac{5}{4}$

Step 8.8

Multiply $- - \frac{5}{4}$ .

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Step 8.8.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{5}{4})$

Step 8.8.2

Multiply $\frac{5}{4}$ by $1$ .

$\frac{5}{4}$