Calculus Examples

${(11 x + 3 y)}^{\frac{1}{3}} = x^{2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(11 x + 3 y)}^{\frac{1}{3}}) = \frac{d}{d x} (x^{2})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 11 x + 3 y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $11 x + 3 y$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [11 x + 3 y]$

Step 2.1.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [11 x + 3 y]$

Step 2.1.3

Replace all occurrences of $u$ with $11 x + 3 y$ .

$\frac{1}{3} {(11 x + 3 y)}^{\frac{1}{3} - 1} \frac{d}{d x} [11 x + 3 y]$

Step 2.2

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

$\frac{1}{3} {(11 x + 3 y)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [11 x + 3 y]$

Step 2.3

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

$\frac{1}{3} {(11 x + 3 y)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [11 x + 3 y]$

Step 2.4

Combine the numerators over the common denominator.

$\frac{1}{3} {(11 x + 3 y)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [11 x + 3 y]$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(11 x + 3 y)}^{\frac{1 - 3}{3}} \frac{d}{d x} [11 x + 3 y]$

Step 2.5.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(11 x + 3 y)}^{\frac{- 2}{3}} \frac{d}{d x} [11 x + 3 y]$

Step 2.6

Combine fractions.

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Step 2.6.1

Move the negative in front of the fraction.

$\frac{1}{3} {(11 x + 3 y)}^{- \frac{2}{3}} \frac{d}{d x} [11 x + 3 y]$

Step 2.6.2

Combine $\frac{1}{3}$ and ${(11 x + 3 y)}^{- \frac{2}{3}}$ .

$\frac{{(11 x + 3 y)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [11 x + 3 y]$

Step 2.6.3

Move ${(11 x + 3 y)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} \frac{d}{d x} [11 x + 3 y]$

Step 2.7

By the Sum Rule, the derivative of $11 x + 3 y$ with respect to $x$ is $\frac{d}{d x} [11 x] + \frac{d}{d x} [3 y]$ .

$\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (\frac{d}{d x} [11 x] + \frac{d}{d x} [3 y])$

Step 2.8

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (11 \frac{d}{d x} [x] + \frac{d}{d x} [3 y])$

Step 2.9

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

$\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (11 \cdot 1 + \frac{d}{d x} [3 y])$

Step 2.10

Multiply $11$ by $1$ .

$\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (11 + \frac{d}{d x} [3 y])$

Step 2.11

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

$\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (11 + 3 \frac{d}{d x} [y])$

Step 2.12

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

$\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (11 + 3 y')$

Step 2.13

Simplify.

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Step 2.13.1

Reorder the factors of $\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (11 + 3 y')$ .

$(11 + 3 y') \frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}}$

Step 2.13.2

Multiply $11 + 3 y'$ by $\frac{1}{3 {(11 x + 3 y)}^{\frac{2}{3}}}$ .

$\frac{11 + 3 y'}{3 {(11 x + 3 y)}^{\frac{2}{3}}}$

Step 3

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

$2 x$

Step 4

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

$\frac{11 + 3 y'}{3 {(11 x + 3 y)}^{\frac{2}{3}}} = 2 x$

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $3 {(11 x + 3 y)}^{\frac{2}{3}}$ .

$\frac{11 + 3 y'}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (3 {(11 x + 3 y)}^{\frac{2}{3}}) = 2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $\frac{11 + 3 y'}{3 {(11 x + 3 y)}^{\frac{2}{3}}} (3 {(11 x + 3 y)}^{\frac{2}{3}})$ .

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Step 5.2.1.1.1

Rewrite using the commutative property of multiplication.

$3 \frac{11 + 3 y'}{3 {(11 x + 3 y)}^{\frac{2}{3}}} {(11 x + 3 y)}^{\frac{2}{3}} = 2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$

Step 5.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

$3 \frac{11 + 3 y'}{3 {(11 x + 3 y)}^{\frac{2}{3}}} {(11 x + 3 y)}^{\frac{2}{3}} = 2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$

Step 5.2.1.1.2.2

Rewrite the expression.

$\frac{11 + 3 y'}{{(11 x + 3 y)}^{\frac{2}{3}}} {(11 x + 3 y)}^{\frac{2}{3}} = 2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$

Step 5.2.1.1.3

Cancel the common factor of ${(11 x + 3 y)}^{\frac{2}{3}}$ .

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Step 5.2.1.1.3.1

Cancel the common factor.

$\frac{11 + 3 y'}{{(11 x + 3 y)}^{\frac{2}{3}}} {(11 x + 3 y)}^{\frac{2}{3}} = 2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$

Step 5.2.1.1.3.2

Rewrite the expression.

$11 + 3 y' = 2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$

Step 5.2.1.1.4

Reorder $11$ and $3 y'$ .

$3 y' + 11 = 2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $2 x (3 {(11 x + 3 y)}^{\frac{2}{3}})$ .

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Step 5.2.2.1.1

Rewrite using the commutative property of multiplication.

$3 y' + 11 = 2 \cdot 3 x {(11 x + 3 y)}^{\frac{2}{3}}$

Step 5.2.2.1.2

Multiply $2$ by $3$ .

$3 y' + 11 = 6 x {(11 x + 3 y)}^{\frac{2}{3}}$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Subtract $11$ from both sides of the equation.

$3 y' = 6 x {(11 x + 3 y)}^{\frac{2}{3}} - 11$

Step 5.3.2

Divide each term in $3 y' = 6 x {(11 x + 3 y)}^{\frac{2}{3}} - 11$ by $3$ and simplify.

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Step 5.3.2.1

Divide each term in $3 y' = 6 x {(11 x + 3 y)}^{\frac{2}{3}} - 11$ by $3$ .

$\frac{3 y'}{3} = \frac{6 x {(11 x + 3 y)}^{\frac{2}{3}}}{3} + \frac{- 11}{3}$

Step 5.3.2.2

Simplify the left side.

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Step 5.3.2.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.2.1.1

Cancel the common factor.

$\frac{3 y'}{3} = \frac{6 x {(11 x + 3 y)}^{\frac{2}{3}}}{3} + \frac{- 11}{3}$

Step 5.3.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{6 x {(11 x + 3 y)}^{\frac{2}{3}}}{3} + \frac{- 11}{3}$

Step 5.3.2.3

Simplify the right side.

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Step 5.3.2.3.1

Simplify each term.

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Step 5.3.2.3.1.1

Factor $3$ out of $6 x {(11 x + 3 y)}^{\frac{2}{3}}$ .

$y' = \frac{3 (2 x {(11 x + 3 y)}^{\frac{2}{3}})}{3} + \frac{- 11}{3}$

Step 5.3.2.3.1.2

Cancel the common factors.

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Step 5.3.2.3.1.2.1

Factor $3$ out of $3$ .

$y' = \frac{3 (2 x {(11 x + 3 y)}^{\frac{2}{3}})}{3 (1)} + \frac{- 11}{3}$

Step 5.3.2.3.1.2.2

Cancel the common factor.

$y' = \frac{3 (2 x {(11 x + 3 y)}^{\frac{2}{3}})}{3 \cdot 1} + \frac{- 11}{3}$

Step 5.3.2.3.1.2.3

Rewrite the expression.

$y' = \frac{2 x {(11 x + 3 y)}^{\frac{2}{3}}}{1} + \frac{- 11}{3}$

Step 5.3.2.3.1.2.4

Divide $2 x {(11 x + 3 y)}^{\frac{2}{3}}$ by $1$ .

$y' = 2 x {(11 x + 3 y)}^{\frac{2}{3}} + \frac{- 11}{3}$

Step 5.3.2.3.1.3

Move the negative in front of the fraction.

$y' = 2 x {(11 x + 3 y)}^{\frac{2}{3}} - \frac{11}{3}$

Step 5.3.2.3.2

To write $2 x {(11 x + 3 y)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y' = 2 x {(11 x + 3 y)}^{\frac{2}{3}} \cdot \frac{3}{3} - \frac{11}{3}$

Step 5.3.2.3.3

Combine $2 x {(11 x + 3 y)}^{\frac{2}{3}}$ and $\frac{3}{3}$ .

$y' = \frac{2 x {(11 x + 3 y)}^{\frac{2}{3}} \cdot 3}{3} - \frac{11}{3}$

Step 5.3.2.3.4

Combine the numerators over the common denominator.

$y' = \frac{2 x {(11 x + 3 y)}^{\frac{2}{3}} \cdot 3 - 11}{3}$

Step 5.3.2.3.5

Multiply $3$ by $2$ .

$y' = \frac{6 x {(11 x + 3 y)}^{\frac{2}{3}} - 11}{3}$

Step 6

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

$\frac{d y}{d x} = \frac{6 x {(11 x + 3 y)}^{\frac{2}{3}} - 11}{3}$