Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x + 3 {(1 - x)}^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3 {(1 - x)}^{\frac{1}{3}}]$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 {(1 - x)}^{\frac{1}{3}}$ with respect to $x$ is $3 \frac{d}{d x} [{(1 - x)}^{\frac{1}{3}}]$ .

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

Step 2.2

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 $g (x) = 1 - x$ .

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $3$ .

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

Step 2.10.2

Subtract $3$ from $1$ .

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

Step 2.11

Move the negative in front of the fraction.

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

Step 2.12

Multiply $- 1$ by $1$ .

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

Step 2.13

Subtract $1$ from $0$ .

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Move $- 1$ to the left of ${(1 - x)}^{- \frac{2}{3}}$ .

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

Step 2.17

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

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

Step 2.18

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

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

Step 2.19

Move the negative in front of the fraction.

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

Step 2.20

Multiply $- 1$ by $3$ .

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

Step 2.21

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

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

Step 2.22

Factor $3$ out of $- 3$ .

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

Step 2.23

Cancel the common factors.

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Step 2.23.1

Factor $3$ out of $3 {(1 - x)}^{\frac{2}{3}}$ .

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

Step 2.23.2

Cancel the common factor.

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

Step 2.23.3

Rewrite the expression.

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

Step 2.24

Move the negative in front of the fraction.

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