Calculus Examples

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

Step 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 $g (x) = 1 - x$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 1$ by $3$ .

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

Step 5.2

Subtract $3$ from $1$ .

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

Step 6

Combine fractions.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Combine fractions.

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Step 12.1

Multiply $- 1$ by $1$ .

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

Step 12.2

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

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

Step 12.3

Move the negative in front of the fraction.

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