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.

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

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Apply basic rules of exponents.

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Step 4.1.1

Rewrite $\frac{1}{{(x - 1)}^{\frac{1}{3}}}$ as ${({(x - 1)}^{\frac{1}{3}})}^{- 1}$ .

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Move the negative in front of the fraction.

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

Step 4.2

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 $g (x) = x - 1$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $3$ .

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

Step 4.6.2

Subtract $3$ from $- 1$ .

$- \frac{1}{3} {(x - 1)}^{\frac{- 4}{3}} \frac{d}{d x} [x - 1]$

Step 4.7

Combine fractions.

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Step 4.7.1

Move the negative in front of the fraction.

$- \frac{1}{3} {(x - 1)}^{- \frac{4}{3}} \frac{d}{d x} [x - 1]$

Step 4.7.2

Combine ${(x - 1)}^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

$- \frac{{(x - 1)}^{- \frac{4}{3}}}{3} \frac{d}{d x} [x - 1]$

Step 4.7.3

Move ${(x - 1)}^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{3 {(x - 1)}^{\frac{4}{3}}} \frac{d}{d x} [x - 1]$

Step 4.8

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

$- \frac{1}{3 {(x - 1)}^{\frac{4}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 4.9

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

$- \frac{1}{3 {(x - 1)}^{\frac{4}{3}}} (1 + \frac{d}{d x} [- 1])$

Step 4.10

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

$- \frac{1}{3 {(x - 1)}^{\frac{4}{3}}} (1 + 0)$

Step 4.11

Simplify the expression.

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Step 4.11.1

Add $1$ and $0$ .

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

Step 4.11.2

Multiply $- 1$ by $1$ .

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

Step 5

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

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

Step 6

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

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