Calculus Examples

$y = \frac{1}{\sqrt[3]{x - 1}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 1}$ as ${(x - 1)}^{\frac{1}{3}}$ .

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

Step 2

Differentiate both sides of the equation.

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

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 1$ and $g (y) = {(x - 1)}^{\frac{1}{3}}$ .

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

Step 4.2

Differentiate using the Constant Rule.

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Step 4.2.1

Multiply $- 1$ by $1$ .

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

Step 4.2.2

Multiply the exponents in ${({(x - 1)}^{\frac{1}{3}})}^{2}$ .

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Step 4.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2.2

Combine $\frac{1}{3}$ and $2$ .

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

Step 4.2.3

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

$\frac{{(x - 1)}^{\frac{1}{3}} \cdot 0 - \frac{d}{d y} [{(x - 1)}^{\frac{1}{3}}]}{{(x - 1)}^{\frac{2}{3}}}$

Step 4.2.4

Simplify the expression.

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Step 4.2.4.1

Multiply ${(x - 1)}^{\frac{1}{3}}$ by $0$ .

$\frac{0 - \frac{d}{d y} [{(x - 1)}^{\frac{1}{3}}]}{{(x - 1)}^{\frac{2}{3}}}$

Step 4.2.4.2

Subtract $\frac{d}{d y} [{(x - 1)}^{\frac{1}{3}}]$ from $0$ .

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

Step 4.2.4.3

Move the negative in front of the fraction.

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

Step 4.3

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 $g (y) = x - 1$ .

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Combine the numerators over the common denominator.

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

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $3$ .

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

Step 4.7.2

Subtract $3$ from $1$ .

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

Step 4.8

Combine fractions.

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Step 4.8.1

Move the negative in front of the fraction.

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

Step 4.8.2

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

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

Step 4.8.3

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

Combine fractions.

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Step 4.12.1

Add $x'$ and $0$ .

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

Step 4.12.2

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

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

Step 4.13

Rewrite $\frac{\frac{x'}{3 {(x - 1)}^{\frac{2}{3}}}}{{(x - 1)}^{\frac{2}{3}}}$ as a product.

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

Step 4.14

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

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

Step 4.15

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

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Step 4.15.1

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

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

Step 4.15.2

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

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

Step 4.15.3

Combine the numerators over the common denominator.

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

Step 4.15.4

Add $2$ and $2$ .

$- \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}}$

Step 5

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

$1 = - \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}}$

Step 6

Solve for $x'$ .

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Step 6.1

Rewrite the equation as $- \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} = 1$ .

$- \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} = 1$

Step 6.2

Divide each term in $- \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} = 1$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} = 1$ by $- 1$ .

$\frac{- \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}}}{- 1} = \frac{1}{- 1}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x'}{3 {(x - 1)}^{\frac{4}{3}}}}{1} = \frac{1}{- 1}$

Step 6.2.2.2

Divide $\frac{x'}{3 {(x - 1)}^{\frac{4}{3}}}$ by $1$ .

$\frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} = \frac{1}{- 1}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $1$ by $- 1$ .

$\frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} = - 1$

Step 6.3

Multiply both sides by $3 {(x - 1)}^{\frac{4}{3}}$ .

$\frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} (3 {(x - 1)}^{\frac{4}{3}}) = - (3 {(x - 1)}^{\frac{4}{3}})$

Step 6.4

Simplify.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Simplify $\frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} (3 {(x - 1)}^{\frac{4}{3}})$ .

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Step 6.4.1.1.1

Rewrite using the commutative property of multiplication.

$3 \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} {(x - 1)}^{\frac{4}{3}} = - (3 {(x - 1)}^{\frac{4}{3}})$

Step 6.4.1.1.2

Cancel the common factor of $3$ .

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Step 6.4.1.1.2.1

Cancel the common factor.

$3 \frac{x'}{3 {(x - 1)}^{\frac{4}{3}}} {(x - 1)}^{\frac{4}{3}} = - (3 {(x - 1)}^{\frac{4}{3}})$

Step 6.4.1.1.2.2

Rewrite the expression.

$\frac{x'}{{(x - 1)}^{\frac{4}{3}}} {(x - 1)}^{\frac{4}{3}} = - (3 {(x - 1)}^{\frac{4}{3}})$

Step 6.4.1.1.3

Cancel the common factor of ${(x - 1)}^{\frac{4}{3}}$ .

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Step 6.4.1.1.3.1

Cancel the common factor.

$\frac{x'}{{(x - 1)}^{\frac{4}{3}}} {(x - 1)}^{\frac{4}{3}} = - (3 {(x - 1)}^{\frac{4}{3}})$

Step 6.4.1.1.3.2

Rewrite the expression.

$x' = - (3 {(x - 1)}^{\frac{4}{3}})$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Multiply $3$ by $- 1$ .

$x' = - 3 {(x - 1)}^{\frac{4}{3}}$

Step 7

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

$\frac{d x}{d y} = - 3 {(x - 1)}^{\frac{4}{3}}$