Calculus Examples

$y = {(5 x - 1)}^{\frac{1}{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} ({(5 x - 1)}^{\frac{1}{3}})$

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 $f (y) = y^{\frac{1}{3}}$ and $g (y) = 5 x - 1$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $5 x - 1$ .

$\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d y} [5 x - 1]$

Step 3.1.2

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

$\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d y} [5 x - 1]$

Step 3.1.3

Replace all occurrences of $u$ with $5 x - 1$ .

$\frac{1}{3} {(5 x - 1)}^{\frac{1}{3} - 1} \frac{d}{d y} [5 x - 1]$

Step 3.2

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

$\frac{1}{3} {(5 x - 1)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d y} [5 x - 1]$

Step 3.3

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

$\frac{1}{3} {(5 x - 1)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d y} [5 x - 1]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{1}{3} {(5 x - 1)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d y} [5 x - 1]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(5 x - 1)}^{\frac{1 - 3}{3}} \frac{d}{d y} [5 x - 1]$

Step 3.5.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(5 x - 1)}^{\frac{- 2}{3}} \frac{d}{d y} [5 x - 1]$

Step 3.6

Combine fractions.

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Step 3.6.1

Move the negative in front of the fraction.

$\frac{1}{3} {(5 x - 1)}^{- \frac{2}{3}} \frac{d}{d y} [5 x - 1]$

Step 3.6.2

Combine $\frac{1}{3}$ and ${(5 x - 1)}^{- \frac{2}{3}}$ .

$\frac{{(5 x - 1)}^{- \frac{2}{3}}}{3} \frac{d}{d y} [5 x - 1]$

Step 3.6.3

Move ${(5 x - 1)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(5 x - 1)}^{\frac{2}{3}}} \frac{d}{d y} [5 x - 1]$

Step 3.7

By the Sum Rule, the derivative of $5 x - 1$ with respect to $y$ is $\frac{d}{d y} [5 x] + \frac{d}{d y} [- 1]$ .

$\frac{1}{3 {(5 x - 1)}^{\frac{2}{3}}} (\frac{d}{d y} [5 x] + \frac{d}{d y} [- 1])$

Step 3.8

Since $5$ is constant with respect to $y$ , the derivative of $5 x$ with respect to $y$ is $5 \frac{d}{d y} [x]$ .

$\frac{1}{3 {(5 x - 1)}^{\frac{2}{3}}} (5 \frac{d}{d y} [x] + \frac{d}{d y} [- 1])$

Step 3.9

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

$\frac{1}{3 {(5 x - 1)}^{\frac{2}{3}}} (5 x' + \frac{d}{d y} [- 1])$

Step 3.10

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

$\frac{1}{3 {(5 x - 1)}^{\frac{2}{3}}} (5 x' + 0)$

Step 3.11

Combine fractions.

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Step 3.11.1

Add $5 x'$ and $0$ .

$\frac{1}{3 {(5 x - 1)}^{\frac{2}{3}}} (5 x')$

Step 3.11.2

Combine $5$ and $\frac{1}{3 {(5 x - 1)}^{\frac{2}{3}}}$ .

$\frac{5}{3 {(5 x - 1)}^{\frac{2}{3}}} x'$

Step 3.11.3

Combine $\frac{5}{3 {(5 x - 1)}^{\frac{2}{3}}}$ and $x'$ .

$\frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}}$

Step 4

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

$1 = \frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}} = 1$ .

$\frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}} = 1$

Step 5.2

Multiply both sides by $3 {(5 x - 1)}^{\frac{2}{3}}$ .

$\frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}} (3 {(5 x - 1)}^{\frac{2}{3}}) = 1 (3 {(5 x - 1)}^{\frac{2}{3}})$

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 $\frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}} (3 {(5 x - 1)}^{\frac{2}{3}})$ .

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Step 5.3.1.1.1

Rewrite using the commutative property of multiplication.

$3 \frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}} {(5 x - 1)}^{\frac{2}{3}} = 1 (3 {(5 x - 1)}^{\frac{2}{3}})$

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.

$3 \frac{5 x'}{3 {(5 x - 1)}^{\frac{2}{3}}} {(5 x - 1)}^{\frac{2}{3}} = 1 (3 {(5 x - 1)}^{\frac{2}{3}})$

Step 5.3.1.1.2.2

Rewrite the expression.

$\frac{5 x'}{{(5 x - 1)}^{\frac{2}{3}}} {(5 x - 1)}^{\frac{2}{3}} = 1 (3 {(5 x - 1)}^{\frac{2}{3}})$

Step 5.3.1.1.3

Cancel the common factor of ${(5 x - 1)}^{\frac{2}{3}}$ .

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Step 5.3.1.1.3.1

Cancel the common factor.

$\frac{5 x'}{{(5 x - 1)}^{\frac{2}{3}}} {(5 x - 1)}^{\frac{2}{3}} = 1 (3 {(5 x - 1)}^{\frac{2}{3}})$

Step 5.3.1.1.3.2

Rewrite the expression.

$5 x' = 1 (3 {(5 x - 1)}^{\frac{2}{3}})$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $3 {(5 x - 1)}^{\frac{2}{3}}$ by $1$ .

$5 x' = 3 {(5 x - 1)}^{\frac{2}{3}}$

Step 5.4

Divide each term in $5 x' = 3 {(5 x - 1)}^{\frac{2}{3}}$ by $5$ and simplify.

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Step 5.4.1

Divide each term in $5 x' = 3 {(5 x - 1)}^{\frac{2}{3}}$ by $5$ .

$\frac{5 x'}{5} = \frac{3 {(5 x - 1)}^{\frac{2}{3}}}{5}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $5$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{5 x'}{5} = \frac{3 {(5 x - 1)}^{\frac{2}{3}}}{5}$

Step 5.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{3 {(5 x - 1)}^{\frac{2}{3}}}{5}$

Step 6

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

$\frac{d x}{d y} = \frac{3 {(5 x - 1)}^{\frac{2}{3}}}{5}$