Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $x + 3 y^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3 y^{\frac{1}{3}}]$ .

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = y$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

$1 + 3 (\frac{1}{3} y^{\frac{1}{3} - 1} y')$

Step 2.2.4

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

$1 + 3 (\frac{1}{3} y^{\frac{1}{3} - 1 \cdot \frac{3}{3}} y')$

Step 2.2.5

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

$1 + 3 (\frac{1}{3} y^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} y')$

Step 2.2.6

Combine the numerators over the common denominator.

$1 + 3 (\frac{1}{3} y^{\frac{1 - 1 \cdot 3}{3}} y')$

Step 2.2.7

Simplify the numerator.

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Step 2.2.7.1

Multiply $- 1$ by $3$ .

$1 + 3 (\frac{1}{3} y^{\frac{1 - 3}{3}} y')$

Step 2.2.7.2

Subtract $3$ from $1$ .

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

Step 2.2.8

Move the negative in front of the fraction.

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

Step 2.2.9

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

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

Step 2.2.10

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

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

Step 2.2.11

Move $y^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2.12

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

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

Step 2.2.13

Cancel the common factor.

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

Step 2.2.14

Rewrite the expression.

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

Step 3

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

$y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $1$ from both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

$y' (\frac{1}{y^{\frac{2}{3}}}) + y' \cdot - 1 = - 1$

Step 5.2.3

Factor $y'$ out of $y' \frac{1}{y^{\frac{2}{3}}} + y' \cdot - 1$ .

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

Step 5.3

Rewrite $1$ as $1^{2}$ .

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

Step 5.4

Rewrite $y^{\frac{2}{3}}$ as ${(y^{\frac{1}{3}})}^{2}$ .

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

Step 5.5

Rewrite $\frac{1^{2}}{{(y^{\frac{1}{3}})}^{2}}$ as ${(\frac{1}{y^{\frac{1}{3}}})}^{2}$ .

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

Step 5.6

Rewrite $1$ as $1^{2}$ .

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

Step 5.7

Factor.

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Step 5.7.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{y^{\frac{1}{3}}}$ and $b = 1$ .

$y' ((\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)) = - 1$

Step 5.7.2

Remove unnecessary parentheses.

$y' (\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1) = - 1$

Step 5.8

Divide each term in $y' (\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1) = - 1$ by $(\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)$ and simplify.

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Step 5.8.1

Divide each term in $y' (\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1) = - 1$ by $(\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)$ .

$\frac{y' (\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)}{(\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)} = \frac{- 1}{(\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)}$

Step 5.8.2

Simplify the left side.

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Step 5.8.2.1

Cancel the common factor.

Step 5.8.2.2

Rewrite the expression.

$\frac{y' (\frac{1}{y^{\frac{1}{3}}} - 1)}{\frac{1}{y^{\frac{1}{3}}} - 1} = \frac{- 1}{(\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)}$

Step 5.8.2.3

Cancel the common factor.

$\frac{y' (\frac{1}{y^{\frac{1}{3}}} - 1)}{\frac{1}{y^{\frac{1}{3}}} - 1} = \frac{- 1}{(\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)}$

Step 5.8.2.4

Divide $y'$ by $1$ .

$y' = \frac{- 1}{(\frac{1}{y^{\frac{1}{3}}} + 1) (\frac{1}{y^{\frac{1}{3}}} - 1)}$

Step 5.8.3

Simplify the right side.

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Step 5.8.3.1

Simplify the denominator.

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Step 5.8.3.1.1

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

$y' = \frac{- 1}{(\frac{1}{y^{\frac{1}{3}}} + \frac{y^{\frac{1}{3}}}{y^{\frac{1}{3}}}) (\frac{1}{y^{\frac{1}{3}}} - 1)}$

Step 5.8.3.1.2

Combine the numerators over the common denominator.

$y' = \frac{- 1}{\frac{1 + y^{\frac{1}{3}}}{y^{\frac{1}{3}}} (\frac{1}{y^{\frac{1}{3}}} - 1)}$

Step 5.8.3.1.3

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

$y' = \frac{- 1}{\frac{1 + y^{\frac{1}{3}}}{y^{\frac{1}{3}}} (\frac{1}{y^{\frac{1}{3}}} - 1 \cdot \frac{y^{\frac{1}{3}}}{y^{\frac{1}{3}}})}$

Step 5.8.3.1.4

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

$y' = \frac{- 1}{\frac{1 + y^{\frac{1}{3}}}{y^{\frac{1}{3}}} (\frac{1}{y^{\frac{1}{3}}} + \frac{- y^{\frac{1}{3}}}{y^{\frac{1}{3}}})}$

Step 5.8.3.1.5

Combine the numerators over the common denominator.

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

Step 5.8.3.2

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

$y' = \frac{- 1}{\frac{(1 + y^{\frac{1}{3}}) (1 - y^{\frac{1}{3}})}{y^{\frac{1}{3}} y^{\frac{1}{3}}}}$

Step 5.8.3.3

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

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Step 5.8.3.3.1

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

$y' = \frac{- 1}{\frac{(1 + y^{\frac{1}{3}}) (1 - y^{\frac{1}{3}})}{y^{\frac{1}{3} + \frac{1}{3}}}}$

Step 5.8.3.3.2

Combine the numerators over the common denominator.

$y' = \frac{- 1}{\frac{(1 + y^{\frac{1}{3}}) (1 - y^{\frac{1}{3}})}{y^{\frac{1 + 1}{3}}}}$

Step 5.8.3.3.3

Add $1$ and $1$ .

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

Step 5.8.3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

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

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