Calculus Examples

$x^{3} + y^{3} = 6 x y$

Step 1

Set each solution of $x^{3} + y^{3}$ as a function of $x$ .

$y = 6 x y \to f (x) = 6 x y$

Step 2

Because the $y$ variable in the equation $x^{3} + y^{3} = 6 x y$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 2.1

Differentiate both sides of the equation.

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

Step 2.2

Differentiate the left side of the equation.

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Step 2.2.1

Differentiate.

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

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

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

Step 2.2.2.1.3

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

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

Step 2.2.2.2

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

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

Step 2.3

Differentiate the right side of the equation.

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Step 2.3.1

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

$6 \frac{d}{d x} [x y]$

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 $g (x) = y$ .

$6 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 2.3.3

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

$6 (x y' + y \frac{d}{d x} [x])$

Step 2.3.4

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

$6 (x y' + y \cdot 1)$

Step 2.3.5

Multiply $y$ by $1$ .

$6 (x y' + y)$

Step 2.3.6

Apply the distributive property.

$6 x y' + 6 y$

Step 2.4

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

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

Step 2.5

Solve for $y'$ .

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Step 2.5.1

Subtract $6 x y'$ from both sides of the equation.

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

Step 2.5.2

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

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

Step 2.5.3

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

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Step 2.5.3.1

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

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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

Step 2.5.4

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

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Step 2.5.4.1

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

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

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $3$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

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

Step 2.5.4.2.1.2

Rewrite the expression.

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

Step 2.5.4.2.2

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

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Step 2.5.4.2.2.1

Cancel the common factor.

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

Step 2.5.4.2.2.2

Divide $y'$ by $1$ .

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

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Simplify each term.

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Step 2.5.4.3.1.1

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

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Step 2.5.4.3.1.1.1

Factor $3$ out of $6 y$ .

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

Step 2.5.4.3.1.1.2

Cancel the common factors.

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Step 2.5.4.3.1.1.2.1

Cancel the common factor.

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

Step 2.5.4.3.1.1.2.2

Rewrite the expression.

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

Step 2.5.4.3.1.2

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

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Step 2.5.4.3.1.2.1

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

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

Step 2.5.4.3.1.2.2

Cancel the common factors.

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Step 2.5.4.3.1.2.2.1

Cancel the common factor.

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

Step 2.5.4.3.1.2.2.2

Rewrite the expression.

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

Step 2.5.4.3.1.3

Move the negative in front of the fraction.

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

Step 2.5.4.3.2

Combine the numerators over the common denominator.

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

Step 2.6

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

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

Step 3

Set the derivative equal to $0$ then solve the equation $\frac{2 y - x^{2}}{y^{2} - 2 x} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$2 y - x^{2} = 0$

Step 3.2

Solve the equation for $x$ .

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Step 3.2.1

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

$- x^{2} = - 2 y$

Step 3.2.2

Divide each term in $- x^{2} = - 2 y$ by $- 1$ and simplify.

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Step 3.2.2.1

Divide each term in $- x^{2} = - 2 y$ by $- 1$ .

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2.2

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

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

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative one from the denominator of $\frac{- 2 y}{- 1}$ .

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

Step 3.2.2.3.2

Rewrite $- 1 \cdot (- 2 y)$ as $- (- 2 y)$ .

$x^{2} = - (- 2 y)$

Step 3.2.2.3.3

Multiply $- 2$ by $- 1$ .

$x^{2} = 2 y$

Step 3.2.3

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

$x = \pm \sqrt{2 y}$

Step 3.2.4

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

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Step 3.2.4.1

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

$x = \sqrt{2 y}$

Step 3.2.4.2

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

$x = - \sqrt{2 y}$

Step 3.2.4.3

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

$x = \sqrt{2 y}$

$x = - \sqrt{2 y}$

$x = \sqrt{2 y}$

$x = - \sqrt{2 y}$

$x = \sqrt{2 y}$

$x = - \sqrt{2 y}$

$x = \sqrt{2 y}$

$x = - \sqrt{2 y}$

Step 4

Solve the function $f (x) = 6 x y$ at $x = \sqrt{2 y}$ .

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Step 4.1

Replace the variable $x$ with $\sqrt{2 y}$ in the expression.

$f (\sqrt{2 y}) = 6 (\sqrt{2 y}) \cdot y$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $6 \sqrt{2 y}$ by $y$ .

$f (\sqrt{2 y}) = 6 \sqrt{2 y} y$

Step 4.2.2

The final answer is $6 \sqrt{2 y} y$ .

$6 \sqrt{2 y} y$

Step 5

Solve the function $f (x) = 6 x y$ at $x = - \sqrt{2 y}$ .

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Step 5.1

Replace the variable $x$ with $- \sqrt{2 y}$ in the expression.

$f (- \sqrt{2 y}) = 6 (- \sqrt{2 y}) \cdot y$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $- 1$ by $6$ .

$f (- \sqrt{2 y}) = - 6 \sqrt{2 y} y$

Step 5.2.2

The final answer is $- 6 \sqrt{2 y} y$ .

$- 6 \sqrt{2 y} y$

Step 6

The horizontal tangent lines are $y = 6 \sqrt{2 y} y, y = - 6 \sqrt{2 y} y$

$y = 6 \sqrt{2 y} y, y = - 6 \sqrt{2 y} y$

Step 7