Calculus Examples

$x^{3} + y^{3} = 21 x y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} + y^{3}) = \frac{d}{d x} (21 x y)$

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 $x^{3} + y^{3}$ 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 2.1.2

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

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

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Step 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 2.2.1.1

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

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

Step 2.2.1.2

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 2.2.1.3

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

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

Step 2.2.2

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

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

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $21$ is constant with respect to $x$ , the derivative of $21 x y$ with respect to $x$ is $21 \frac{d}{d x} [x y]$ .

$21 \frac{d}{d x} [x y]$

Step 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 $g (x) = y$ .

$21 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 3.3

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

$21 (x y' + y \frac{d}{d x} [x])$

Step 3.4

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

$21 (x y' + y \cdot 1)$

Step 3.5

Multiply $y$ by $1$ .

$21 (x y' + y)$

Step 3.6

Apply the distributive property.

$21 x y' + 21 y$

Step 4

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

$3 x^{2} + 3 y^{2} y' = 21 x y' + 21 y$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $21 x y'$ from both sides of the equation.

$3 x^{2} + 3 y^{2} y' - 21 x y' = 21 y$

Step 5.2

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

$3 y^{2} y' - 21 x y' = 21 y - 3 x^{2}$

Step 5.3

Factor $3 y'$ out of $3 y^{2} y' - 21 x y'$ .

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Step 5.3.1

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

$3 y' y^{2} - 21 x y' = 21 y - 3 x^{2}$

Step 5.3.2

Factor $3 y'$ out of $- 21 x y'$ .

$3 y' y^{2} + 3 y' (- 7 x) = 21 y - 3 x^{2}$

Step 5.3.3

Factor $3 y'$ out of $3 y' y^{2} + 3 y' (- 7 x)$ .

$3 y' (y^{2} - 7 x) = 21 y - 3 x^{2}$

Step 5.4

Divide each term in $3 y' (y^{2} - 7 x) = 21 y - 3 x^{2}$ by $3 (y^{2} - 7 x)$ and simplify.

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Step 5.4.1

Divide each term in $3 y' (y^{2} - 7 x) = 21 y - 3 x^{2}$ by $3 (y^{2} - 7 x)$ .

$\frac{3 y' (y^{2} - 7 x)}{3 (y^{2} - 7 x)} = \frac{21 y}{3 (y^{2} - 7 x)} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{3 y' (y^{2} - 7 x)}{3 (y^{2} - 7 x)} = \frac{21 y}{3 (y^{2} - 7 x)} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.2.1.2

Rewrite the expression.

$\frac{y' (y^{2} - 7 x)}{y^{2} - 7 x} = \frac{21 y}{3 (y^{2} - 7 x)} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.2.2

Cancel the common factor of $y^{2} - 7 x$ .

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Step 5.4.2.2.1

Cancel the common factor.

$\frac{y' (y^{2} - 7 x)}{y^{2} - 7 x} = \frac{21 y}{3 (y^{2} - 7 x)} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{21 y}{3 (y^{2} - 7 x)} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Cancel the common factor of $21$ and $3$ .

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Step 5.4.3.1.1.1

Factor $3$ out of $21 y$ .

$y' = \frac{3 (7 y)}{3 (y^{2} - 7 x)} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.3.1.1.2

Cancel the common factors.

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Step 5.4.3.1.1.2.1

Cancel the common factor.

$y' = \frac{3 (7 y)}{3 (y^{2} - 7 x)} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.3.1.1.2.2

Rewrite the expression.

$y' = \frac{7 y}{y^{2} - 7 x} + \frac{- 3 x^{2}}{3 (y^{2} - 7 x)}$

Step 5.4.3.1.2

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

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Step 5.4.3.1.2.1

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

$y' = \frac{7 y}{y^{2} - 7 x} + \frac{3 (- x^{2})}{3 (y^{2} - 7 x)}$

Step 5.4.3.1.2.2

Cancel the common factors.

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Step 5.4.3.1.2.2.1

Cancel the common factor.

$y' = \frac{7 y}{y^{2} - 7 x} + \frac{3 (- x^{2})}{3 (y^{2} - 7 x)}$

Step 5.4.3.1.2.2.2

Rewrite the expression.

$y' = \frac{7 y}{y^{2} - 7 x} + \frac{- x^{2}}{y^{2} - 7 x}$

Step 5.4.3.1.3

Move the negative in front of the fraction.

$y' = \frac{7 y}{y^{2} - 7 x} - \frac{x^{2}}{y^{2} - 7 x}$

Step 5.4.3.2

Combine the numerators over the common denominator.

$y' = \frac{7 y - x^{2}}{y^{2} - 7 x}$

Step 6

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

$\frac{d y}{d x} = \frac{7 y - x^{2}}{y^{2} - 7 x}$

Step 7

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

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Step 7.1

Set the numerator equal to zero.

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

Step 7.2

Solve the equation for $x$ .

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Step 7.2.1

Subtract $7 y$ from both sides of the equation.

$- x^{2} = - 7 y$

Step 7.2.2

Divide each term in $- x^{2} = - 7 y$ by $- 1$ and simplify.

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Step 7.2.2.1

Divide each term in $- x^{2} = - 7 y$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 7 y}{- 1}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 7 y}{- 1}$

Step 7.2.2.2.2

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

$x^{2} = \frac{- 7 y}{- 1}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Move the negative one from the denominator of $\frac{- 7 y}{- 1}$ .

$x^{2} = - 1 \cdot (- 7 y)$

Step 7.2.2.3.2

Rewrite $- 1 \cdot (- 7 y)$ as $- (- 7 y)$ .

$x^{2} = - (- 7 y)$

Step 7.2.2.3.3

Multiply $- 7$ by $- 1$ .

$x^{2} = 7 y$

Step 7.2.3

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

$x = \pm \sqrt{7 y}$

Step 7.2.4

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

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Step 7.2.4.1

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

$x = \sqrt{7 y}$

Step 7.2.4.2

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

$x = - \sqrt{7 y}$

Step 7.2.4.3

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

$x = \sqrt{7 y}$

$x = - \sqrt{7 y}$

$x = \sqrt{7 y}$

$x = - \sqrt{7 y}$

$x = \sqrt{7 y}$

$x = - \sqrt{7 y}$

$x = \sqrt{7 y}$

$x = - \sqrt{7 y}$

Step 8

Solve for $y$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite ${\sqrt{7 y}}^{3}$ as $\sqrt{{(7 y)}^{3}}$ .

$\sqrt{{(7 y)}^{3}} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.2

Apply the product rule to $7 y$ .

$\sqrt{7^{3} y^{3}} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.3

Raise $7$ to the power of $3$ .

$\sqrt{343 y^{3}} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.4

Rewrite $343 y^{3}$ as ${(7 y)}^{2} \cdot (7 y)$ .

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Step 8.1.4.1

Factor $49$ out of $343$ .

$\sqrt{49 (7) y^{3}} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.4.2

Rewrite $49$ as $7^{2}$ .

$\sqrt{7^{2} \cdot 7 y^{3}} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.4.3

Factor out $y^{2}$ .

$\sqrt{7^{2} \cdot 7 (y^{2} y)} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.4.4

Move $7$ .

$\sqrt{7^{2} y^{2} \cdot 7 y} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.4.5

Rewrite $7^{2} y^{2}$ as ${(7 y)}^{2}$ .

$\sqrt{{(7 y)}^{2} \cdot 7 y} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.4.6

Add parentheses.

$\sqrt{{(7 y)}^{2} \cdot (7 y)} + y^{3} = 21 \sqrt{7 y} y$

Step 8.1.5

Pull terms out from under the radical.

$7 y \sqrt{7 y} + y^{3} = 21 \sqrt{7 y} y$

Step 8.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 y}$ as ${(7 y)}^{\frac{1}{2}}$ .

$7 y {(7 y)}^{\frac{1}{2}} + y^{3} = 21 \sqrt{7 y} y$

Step 8.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 y}$ as ${(7 y)}^{\frac{1}{2}}$ .

$7 y {(7 y)}^{\frac{1}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4

Simplify each term.

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Step 8.4.1

Apply the product rule to $7 y$ .

$7 y (7^{\frac{1}{2}} y^{\frac{1}{2}}) + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.2

Rewrite using the commutative property of multiplication.

$7 \cdot 7^{\frac{1}{2}} y \cdot y^{\frac{1}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.3

Multiply $7$ by $7^{\frac{1}{2}}$ by adding the exponents.

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Step 8.4.3.1

Multiply $7$ by $7^{\frac{1}{2}}$ .

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Step 8.4.3.1.1

Raise $7$ to the power of $1$ .

$7^{1} \cdot 7^{\frac{1}{2}} y \cdot y^{\frac{1}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.3.1.2

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

$7^{1 + \frac{1}{2}} y \cdot y^{\frac{1}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.3.2

Write $1$ as a fraction with a common denominator.

$7^{\frac{2}{2} + \frac{1}{2}} y \cdot y^{\frac{1}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.3.3

Combine the numerators over the common denominator.

$7^{\frac{2 + 1}{2}} y \cdot y^{\frac{1}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.3.4

Add $2$ and $1$ .

$7^{\frac{3}{2}} y \cdot y^{\frac{1}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.4

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

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Step 8.4.4.1

Move $y^{\frac{1}{2}}$ .

$7^{\frac{3}{2}} (y^{\frac{1}{2}} y) + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.4.2

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

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Step 8.4.4.2.1

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

$7^{\frac{3}{2}} (y^{\frac{1}{2}} y^{1}) + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.4.2.2

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

$7^{\frac{3}{2}} y^{\frac{1}{2} + 1} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.4.3

Write $1$ as a fraction with a common denominator.

$7^{\frac{3}{2}} y^{\frac{1}{2} + \frac{2}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.4.4

Combine the numerators over the common denominator.

$7^{\frac{3}{2}} y^{\frac{1 + 2}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.4.4.5

Add $1$ and $2$ .

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 {(7 y)}^{\frac{1}{2}} y$

Step 8.5

Simplify $21 {(7 y)}^{\frac{1}{2}} y$ .

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Step 8.5.1

Apply the product rule to $7 y$ .

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 (7^{\frac{1}{2}} y^{\frac{1}{2}}) y$

Step 8.5.2

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

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Step 8.5.2.1

Move $y$ .

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 (7^{\frac{1}{2}} (y \cdot y^{\frac{1}{2}}))$

Step 8.5.2.2

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

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Step 8.5.2.2.1

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

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 (7^{\frac{1}{2}} (y^{1} y^{\frac{1}{2}}))$

Step 8.5.2.2.2

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

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 (7^{\frac{1}{2}} y^{1 + \frac{1}{2}})$

Step 8.5.2.3

Write $1$ as a fraction with a common denominator.

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 (7^{\frac{1}{2}} y^{\frac{2}{2} + \frac{1}{2}})$

Step 8.5.2.4

Combine the numerators over the common denominator.

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 (7^{\frac{1}{2}} y^{\frac{2 + 1}{2}})$

Step 8.5.2.5

Add $2$ and $1$ .

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 (7^{\frac{1}{2}} y^{\frac{3}{2}})$

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} = 21 \cdot 7^{\frac{1}{2}} y^{\frac{3}{2}}$

Step 8.6

Subtract $21 \cdot 7^{\frac{1}{2}} y^{\frac{3}{2}}$ from both sides of the equation.

$7^{\frac{3}{2}} y^{\frac{3}{2}} + y^{3} - 21 \cdot 7^{\frac{1}{2}} y^{\frac{3}{2}} = 0$

Step 8.7

Find a common factor $y^{\frac{3}{2}}$ that is present in each term.

$7^{\frac{3}{2}} y^{\frac{3}{2}} {(y^{\frac{3}{2}})}^{2} - 21 \cdot (7^{\frac{1}{2}} y^{\frac{3}{2}})$

Step 8.8

Substitute $u$ for $y^{\frac{3}{2}}$ .

$7^{\frac{3}{2}} (u) {(u)}^{2} - 21 \cdot (7^{\frac{1}{2}} (u)) = 0$

Step 8.9

Solve for $u$ .

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Step 8.9.1

Simplify $7^{\frac{3}{2}} (u) {(u)}^{2} - 21 \cdot (7^{\frac{1}{2}} (u))$ .

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Step 8.9.1.1

Multiply $u$ by ${(u)}^{2}$ by adding the exponents.

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Step 8.9.1.1.1

Move ${(u)}^{2}$ .

$7^{\frac{3}{2}} ({(u)}^{2} u) - 21 \cdot (7^{\frac{1}{2}} (u)) = 0$

Step 8.9.1.1.2

Multiply ${(u)}^{2}$ by $u$ .

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Step 8.9.1.1.2.1

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

$7^{\frac{3}{2}} ({(u)}^{2} u^{1}) - 21 \cdot (7^{\frac{1}{2}} (u)) = 0$

Step 8.9.1.1.2.2

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

$7^{\frac{3}{2}} u^{2 + 1} - 21 \cdot (7^{\frac{1}{2}} (u)) = 0$

Step 8.9.1.1.3

Add $2$ and $1$ .

$7^{\frac{3}{2}} u^{3} - 21 \cdot (7^{\frac{1}{2}} (u)) = 0$

$7^{\frac{3}{2}} u^{3} - 21 \cdot 7^{\frac{1}{2}} u = 0$

Step 8.9.1.2

Reorder factors in $7^{\frac{3}{2}} u^{3} - 21 \cdot 7^{\frac{1}{2}} u$ .

$u^{3} \cdot 7^{\frac{3}{2}} - 21 u \cdot 7^{\frac{1}{2}} = 0$

Step 8.9.2

Factor the left side of the equation.

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Step 8.9.2.1

Factor $u \cdot 7^{\frac{1}{2}}$ out of $u^{3} \cdot 7^{\frac{3}{2}} - 21 u \cdot 7^{\frac{1}{2}}$ .

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Step 8.9.2.1.1

Reorder the expression.

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Step 8.9.2.1.1.1

Reorder $u^{3}$ and $7^{\frac{3}{2}}$ .

$7^{\frac{3}{2}} \cdot u^{3} - 21 u \cdot 7^{\frac{1}{2}} = 0$

Step 8.9.2.1.1.2

Move $u$ .

$7^{\frac{3}{2}} \cdot u^{3} - 21 \cdot (7^{\frac{1}{2}} u) = 0$

Step 8.9.2.1.2

Factor $u \cdot 7^{\frac{1}{2}}$ out of $7^{\frac{3}{2}} \cdot u^{3}$ .

$u \cdot (7^{\frac{1}{2}} (7^{\frac{2}{2}} \cdot u^{2})) - 21 \cdot (7^{\frac{1}{2}} u) = 0$

Step 8.9.2.1.3

Factor $u \cdot 7^{\frac{1}{2}}$ out of $- 21 \cdot 7^{\frac{1}{2}} u$ .

$u \cdot (7^{\frac{1}{2}} (7^{\frac{2}{2}} \cdot u^{2})) + u \cdot (7^{\frac{1}{2}} (- 21)) = 0$

Step 8.9.2.1.4

Factor $u \cdot 7^{\frac{1}{2}}$ out of $u \cdot 7^{\frac{1}{2}} (7^{\frac{2}{2}} \cdot u^{2}) + u \cdot 7^{\frac{1}{2}} (- 21)$ .

$u \cdot (7^{\frac{1}{2}} (7^{\frac{2}{2}} \cdot u^{2} - 21)) = 0$

Step 8.9.2.2

Divide $2$ by $2$ .

$u \cdot (7^{\frac{1}{2}} (7 \cdot u^{2} - 21)) = 0$

Step 8.9.2.3

Evaluate the exponent.

$u \cdot (7^{\frac{1}{2}} (7 u^{2} - 21)) = 0$

Step 8.9.2.4

Factor.

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Step 8.9.2.4.1

Factor $7$ out of $7 u^{2} - 21$ .

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Step 8.9.2.4.1.1

Factor $7$ out of $7 u^{2}$ .

$u \cdot (7^{\frac{1}{2}} (7 (u^{2}) - 21)) = 0$

Step 8.9.2.4.1.2

Factor $7$ out of $- 21$ .

$u \cdot (7^{\frac{1}{2}} (7 u^{2} + 7 \cdot - 3)) = 0$

Step 8.9.2.4.1.3

Factor $7$ out of $7 u^{2} + 7 \cdot - 3$ .

$u \cdot (7^{\frac{1}{2}} (7 (u^{2} - 3))) = 0$

Step 8.9.2.4.2

Remove unnecessary parentheses.

$u \cdot 7^{\frac{1}{2}} \cdot (7 (u^{2} - 3)) = 0$

Step 8.9.2.5

Combine exponents.

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Step 8.9.2.5.1

Raise $7$ to the power of $1$ .

$u \cdot ((7 \cdot 7^{\frac{1}{2}}) (u^{2} - 3)) = 0$

Step 8.9.2.5.2

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

$u \cdot (7^{1 + \frac{1}{2}} (u^{2} - 3)) = 0$

Step 8.9.2.5.3

Write $1$ as a fraction with a common denominator.

$u \cdot (7^{\frac{2}{2} + \frac{1}{2}} (u^{2} - 3)) = 0$

Step 8.9.2.5.4

Combine the numerators over the common denominator.

$u \cdot (7^{\frac{2 + 1}{2}} (u^{2} - 3)) = 0$

Step 8.9.2.5.5

Add $2$ and $1$ .

$u \cdot (7^{\frac{3}{2}} (u^{2} - 3)) = 0$

Step 8.9.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$u^{2} - 3 = 0$

Step 8.9.4

Set $u$ equal to $0$ .

$u = 0$

Step 8.9.5

Set $u^{2} - 3$ equal to $0$ and solve for $u$ .

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Step 8.9.5.1

Set $u^{2} - 3$ equal to $0$ .

$u^{2} - 3 = 0$

Step 8.9.5.2

Solve $u^{2} - 3 = 0$ for $u$ .

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Step 8.9.5.2.1

Add $3$ to both sides of the equation.

$u^{2} = 3$

Step 8.9.5.2.2

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

$u = \pm \sqrt{3}$

Step 8.9.5.2.3

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

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Step 8.9.5.2.3.1

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

$u = \sqrt{3}$

Step 8.9.5.2.3.2

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

$u = - \sqrt{3}$

Step 8.9.5.2.3.3

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

$u = \sqrt{3}, - \sqrt{3}$

Step 8.9.6

The final solution is all the values that make $u \cdot (7^{\frac{3}{2}} (u^{2} - 3)) = 0$ true.

$u = 0, \sqrt{3}, - \sqrt{3}$

Step 8.10

Substitute $y$ for $u$ .

$y^{\frac{3}{2}} = 0, \sqrt{3}, - \sqrt{3}$

Step 8.11

Solve for $y^{\frac{3}{2}} = 0$ for $y$ .

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Step 8.11.1

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{3}{2}})}^{\frac{2}{3}} = 0^{\frac{2}{3}}$

Step 8.11.2

Simplify the exponent.

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Step 8.11.2.1

Simplify the left side.

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Step 8.11.2.1.1

Simplify ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 8.11.2.1.1.1

Multiply the exponents in ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 8.11.2.1.1.1.1

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

$y^{\frac{3}{2} \cdot \frac{2}{3}} = 0^{\frac{2}{3}}$

Step 8.11.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 8.11.2.1.1.1.2.1

Cancel the common factor.

$y^{\frac{3}{2} \cdot \frac{2}{3}} = 0^{\frac{2}{3}}$

Step 8.11.2.1.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{2} \cdot 2} = 0^{\frac{2}{3}}$

Step 8.11.2.1.1.1.3

Cancel the common factor of $2$ .

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Step 8.11.2.1.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = 0^{\frac{2}{3}}$

Step 8.11.2.1.1.1.3.2

Rewrite the expression.

$y^{1} = 0^{\frac{2}{3}}$

Step 8.11.2.1.1.2

Simplify.

$y = 0^{\frac{2}{3}}$

Step 8.11.2.2

Simplify the right side.

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Step 8.11.2.2.1

Simplify $0^{\frac{2}{3}}$ .

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Step 8.11.2.2.1.1

Simplify the expression.

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Step 8.11.2.2.1.1.1

Rewrite $0$ as $0^{3}$ .

$y = {(0^{3})}^{\frac{2}{3}}$

Step 8.11.2.2.1.1.2

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

$y = 0^{3 (\frac{2}{3})}$

Step 8.11.2.2.1.2

Cancel the common factor of $3$ .

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Step 8.11.2.2.1.2.1

Cancel the common factor.

$y = 0^{3 (\frac{2}{3})}$

Step 8.11.2.2.1.2.2

Rewrite the expression.

$y = 0^{2}$

Step 8.11.2.2.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0$

Step 8.12

Solve for $y^{\frac{3}{2}} = \sqrt{3}$ for $y$ .

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Step 8.12.1

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

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

Step 8.12.2

Simplify the exponent.

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Step 8.12.2.1

Simplify the left side.

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Step 8.12.2.1.1

Simplify ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 8.12.2.1.1.1

Multiply the exponents in ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 8.12.2.1.1.1.1

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

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

Step 8.12.2.1.1.1.2

Cancel the common factor of $3$ .

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Step 8.12.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.12.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.12.2.1.1.1.3

Cancel the common factor of $2$ .

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Step 8.12.2.1.1.1.3.1

Cancel the common factor.

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

Step 8.12.2.1.1.1.3.2

Rewrite the expression.

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

Step 8.12.2.1.1.2

Simplify.

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

Step 8.12.2.2

Simplify the right side.

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Step 8.12.2.2.1

Rewrite ${\sqrt{3}}^{\frac{2}{3}}$ as $\sqrt[3]{3}$ .

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Step 8.12.2.2.1.1

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

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

Step 8.12.2.2.1.2

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

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

Step 8.12.2.2.1.3

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

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

Step 8.12.2.2.1.4

Multiply $2$ by $3$ .

$y = 3^{\frac{2}{6}}$

Step 8.12.2.2.1.5

Cancel the common factor of $2$ and $6$ .

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Step 8.12.2.2.1.5.1

Factor $2$ out of $2$ .

$y = 3^{\frac{2 (1)}{6}}$

Step 8.12.2.2.1.5.2

Cancel the common factors.

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Step 8.12.2.2.1.5.2.1

Factor $2$ out of $6$ .

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

Step 8.12.2.2.1.5.2.2

Cancel the common factor.

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

Step 8.12.2.2.1.5.2.3

Rewrite the expression.

$y = 3^{\frac{1}{3}}$

Step 8.12.2.2.1.6

Rewrite $3^{\frac{1}{3}}$ as $\sqrt[3]{3}$ .

$y = \sqrt[3]{3}$

Step 8.13

List all of the solutions.

$y = 0, \sqrt[3]{3}, \sqrt[3]{3}$

Step 8.14

Exclude the solutions that do not make ${(\sqrt{7 y})}^{3} + y^{3} = 21 (\sqrt{7 y}) y$ true.

$y = 0$

Step 9

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

$(\sqrt{7 y}, 0)$

$(- \sqrt{7 y}, 0)$

Step 10