Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- x^{\frac{2}{3}} + y^{\frac{2}{3}}) = \frac{d}{d x} (1)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $- x^{\frac{2}{3}} + y^{\frac{2}{3}}$ with respect to $x$ is $\frac{d}{d x} [- x^{\frac{2}{3}}] + \frac{d}{d x} [y^{\frac{2}{3}}]$ .

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

Step 2.2

Evaluate $\frac{d}{d x} [- x^{\frac{2}{3}}]$ .

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{2}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{2}{3}}]$ .

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

Step 2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 2.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.2.4

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 2.2.5

Combine the numerators over the common denominator.

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

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.2.6.2

Subtract $3$ from $2$ .

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

Step 2.2.7

Move the negative in front of the fraction.

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

Step 2.2.8

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$- \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 2.2.9

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [y^{\frac{2}{3}}]$

Step 2.3

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

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Step 2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{2}{3}}$ and $g (x) = y$ .

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Step 2.3.1.1

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

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [y]$

Step 2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{2}{3}$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [y]$

Step 2.3.1.3

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

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{\frac{2}{3} - 1} \frac{d}{d x} [y]$

Step 2.3.2

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

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{\frac{2}{3} - 1} y'$

Step 2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{\frac{2}{3} - 1 \cdot \frac{3}{3}} y'$

Step 2.3.4

Combine $- 1$ and $\frac{3}{3}$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} y'$

Step 2.3.5

Combine the numerators over the common denominator.

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{\frac{2 - 1 \cdot 3}{3}} y'$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $3$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{\frac{2 - 3}{3}} y'$

Step 2.3.6.2

Subtract $3$ from $2$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{\frac{- 1}{3}} y'$

Step 2.3.7

Move the negative in front of the fraction.

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2}{3} y^{- \frac{1}{3}} y'$

Step 2.3.8

Combine $\frac{2}{3}$ and $y^{- \frac{1}{3}}$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2 y^{- \frac{1}{3}}}{3} y'$

Step 2.3.9

Combine $\frac{2 y^{- \frac{1}{3}}}{3}$ and $y'$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2 y^{- \frac{1}{3}} y'}{3}$

Step 2.3.10

Move $y^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{2}{3 x^{\frac{1}{3}}} + \frac{2 y'}{3 y^{\frac{1}{3}}}$

Step 2.4

Reorder terms.

$\frac{2 y'}{3 y^{\frac{1}{3}}} - \frac{2}{3 x^{\frac{1}{3}}}$

Step 3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Add $\frac{2}{3 x^{\frac{1}{3}}}$ to both sides of the equation.

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

Step 5.2

Multiply both sides by $3 y^{\frac{1}{3}}$ .

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

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{2 y'}{3 y^{\frac{1}{3}}} (3 y^{\frac{1}{3}})$ .

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Step 5.3.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.1.2

Cancel the common factor of $3$ .

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Step 5.3.1.1.2.1

Cancel the common factor.

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

Step 5.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.1.1.3

Cancel the common factor of $y^{\frac{1}{3}}$ .

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Step 5.3.1.1.3.1

Cancel the common factor.

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

Step 5.3.1.1.3.2

Rewrite the expression.

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $\frac{2}{3 x^{\frac{1}{3}}} (3 y^{\frac{1}{3}})$ .

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Step 5.3.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.2.1.2

Cancel the common factor of $3$ .

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Step 5.3.2.1.2.1

Cancel the common factor.

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

Step 5.3.2.1.2.2

Rewrite the expression.

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

Step 5.3.2.1.3

Combine $\frac{2}{x^{\frac{1}{3}}}$ and $y^{\frac{1}{3}}$ .

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

Step 5.4

Divide each term in $2 y' = \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ by $2$ and simplify.

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Step 5.4.1

Divide each term in $2 y' = \frac{2 y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ by $2$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4.3.2

Combine.

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

Step 5.4.3.3

Cancel the common factor.

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

Step 5.4.3.4

Rewrite the expression.

$y' = \frac{y^{\frac{1}{3}} \cdot 1}{x^{\frac{1}{3}}}$

Step 5.4.3.5

Multiply $y^{\frac{1}{3}}$ by $1$ .

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

Step 6

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

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