Calculus Examples

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

Step 1

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

$y = 2 x y \to f (x) = 2 x y$

Step 2

Because the $y$ variable in the equation $x^{3} + y^{3} = 2 x y$ 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.

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

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

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

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 $g (x) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $y$ by $1$ .

$2 (x y' + y)$

Step 2.3.6

Apply the distributive property.

$2 x y' + 2 y$

Step 2.4

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

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

Step 2.5

Solve for $y'$ .

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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

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Step 2.5.3.1

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

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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

Step 2.5.4

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

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Step 2.5.4.1

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

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

Step 2.5.4.2

Simplify the left side.

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Step 2.5.4.2.1

Cancel the common factor of $3 y^{2} - 2 x$ .

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Step 2.5.4.2.1.1

Cancel the common factor.

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

Step 2.5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 2.5.4.3

Simplify the right side.

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Step 2.5.4.3.1

Combine the numerators over the common denominator.

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

Step 2.6

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

$\frac{d y}{d x} = \frac{2 y - 3 x^{2}}{3 y^{2} - 2 x}$

Step 3

Set the derivative equal to $0$ then solve the equation $\frac{2 y - 3 x^{2}}{3 y^{2} - 2 x} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$2 y - 3 x^{2} = 0$

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.

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

Step 3.2.2

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

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Step 3.2.2.1

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

$\frac{- 3 x^{2}}{- 3} = \frac{- 2 y}{- 3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{- 3 x^{2}}{- 3} = \frac{- 2 y}{- 3}$

Step 3.2.2.2.1.2

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

$x^{2} = \frac{- 2 y}{- 3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Dividing two negative values results in a positive value.

$x^{2} = \frac{2 y}{3}$

Step 3.2.3

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

$x = \pm \sqrt{\frac{2 y}{3}}$

Step 3.2.4

Simplify $\pm \sqrt{\frac{2 y}{3}}$ .

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Step 3.2.4.1

Rewrite $\sqrt{\frac{2 y}{3}}$ as $\frac{\sqrt{2 y}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2 y}}{\sqrt{3}}$

Step 3.2.4.2

Multiply $\frac{\sqrt{2 y}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2 y}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.2.4.3

Combine and simplify the denominator.

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Step 3.2.4.3.1

Multiply $\frac{\sqrt{2 y}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.2.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.2.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.2.4.3.4

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

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.2.4.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.2.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.2.4.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.2.4.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.2.4.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.2.4.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{3^{1}}$

Step 3.2.4.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2 y} \sqrt{3}}{3}$

Step 3.2.4.4

Simplify the numerator.

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Step 3.2.4.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{2 y \cdot 3}}{3}$

Step 3.2.4.4.2

Multiply $3$ by $2$ .

$x = \pm \frac{\sqrt{6 y}}{3}$

Step 3.2.5

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

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Step 3.2.5.1

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

$x = \frac{\sqrt{6 y}}{3}$

Step 3.2.5.2

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

$x = - \frac{\sqrt{6 y}}{3}$

Step 3.2.5.3

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

$x = \frac{\sqrt{6 y}}{3}$

$x = - \frac{\sqrt{6 y}}{3}$

$x = \frac{\sqrt{6 y}}{3}$

$x = - \frac{\sqrt{6 y}}{3}$

$x = \frac{\sqrt{6 y}}{3}$

$x = - \frac{\sqrt{6 y}}{3}$

$x = \frac{\sqrt{6 y}}{3}$

$x = - \frac{\sqrt{6 y}}{3}$

Step 4

Solve the function $f (x) = 2 x y$ at $x = \frac{\sqrt{6 y}}{3}$ .

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Step 4.1

Replace the variable $x$ with $\frac{\sqrt{6 y}}{3}$ in the expression.

$f (\frac{\sqrt{6 y}}{3}) = 2 (\frac{\sqrt{6 y}}{3}) \cdot y$

Step 4.2

Simplify the result.

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Step 4.2.1

Combine $2$ and $\frac{\sqrt{6 y}}{3}$ .

$f (\frac{\sqrt{6 y}}{3}) = \frac{2 \sqrt{6 y}}{3} \cdot y$

Step 4.2.2

Combine $\frac{2 \sqrt{6 y}}{3}$ and $y$ .

$f (\frac{\sqrt{6 y}}{3}) = \frac{2 \sqrt{6 y} y}{3}$

Step 4.2.3

The final answer is $\frac{2 \sqrt{6 y} y}{3}$ .

$\frac{2 \sqrt{6 y} y}{3}$

Step 5

Solve the function $f (x) = 2 x y$ at $x = - \frac{\sqrt{6 y}}{3}$ .

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Step 5.1

Replace the variable $x$ with $- \frac{\sqrt{6 y}}{3}$ in the expression.

$f (- \frac{\sqrt{6 y}}{3}) = 2 (- \frac{\sqrt{6 y}}{3}) \cdot y$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2 (- \frac{\sqrt{6 y}}{3})$ .

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Step 5.2.1.1

Multiply $- 1$ by $2$ .

$f (- \frac{\sqrt{6 y}}{3}) = - 2 \frac{\sqrt{6 y}}{3} \cdot y$

Step 5.2.1.2

Combine $- 2$ and $\frac{\sqrt{6 y}}{3}$ .

$f (- \frac{\sqrt{6 y}}{3}) = \frac{- 2 \sqrt{6 y}}{3} \cdot y$

Step 5.2.2

Move the negative in front of the fraction.

$f (- \frac{\sqrt{6 y}}{3}) = - \frac{(2) \sqrt{6 y}}{3} \cdot y$

Step 5.2.3

Combine $y$ and $\frac{2 \sqrt{6 y}}{3}$ .

$f (- \frac{\sqrt{6 y}}{3}) = - \frac{y (2 \sqrt{6 y})}{3}$

Step 5.2.4

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

$f (- \frac{\sqrt{6 y}}{3}) = - \frac{2 y \sqrt{6 y}}{3}$

Step 5.2.5

The final answer is $- \frac{2 y \sqrt{6 y}}{3}$ .

$- \frac{2 y \sqrt{6 y}}{3}$

Step 6

The horizontal tangent lines are $y = \frac{2 \sqrt{6 y} y}{3}, y = - \frac{2 y \sqrt{6 y}}{3}$

$y = \frac{2 \sqrt{6 y} y}{3}, y = - \frac{2 y \sqrt{6 y}}{3}$

Step 7