Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $3$ .

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

Step 3.5.2

Subtract $3$ from $2$ .

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

Step 3.6

Move the negative in front of the fraction.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{2 x'}{3 x^{\frac{1}{3}}} = 1$ .

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

Step 5.2

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

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

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

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Step 5.3.1.1.1

Rewrite using the commutative property of multiplication.

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

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.

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

Step 5.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.1.1.3

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

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Step 5.3.1.1.3.1

Cancel the common factor.

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

Step 5.3.1.1.3.2

Rewrite the expression.

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

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

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

Step 5.4

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

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Step 5.4.1

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

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

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.

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

Step 5.4.2.1.2

Divide $x'$ by $1$ .

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

Step 6

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

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