Calculus Examples

$f (x) = 5 x \sqrt[3]{x - 8}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 8}$ as ${(x - 8)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [5 x {(x - 8)}^{\frac{1}{3}}]$

Step 1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x {(x - 8)}^{\frac{1}{3}}$ with respect to $x$ is $5 \frac{d}{d x} [x {(x - 8)}^{\frac{1}{3}}]$ .

$5 \frac{d}{d x} [x {(x - 8)}^{\frac{1}{3}}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(x - 8)}^{\frac{1}{3}}$ .

$5 (x \frac{d}{d x} [{(x - 8)}^{\frac{1}{3}}] + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 3

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 $g (x) = x - 8$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x - 8$ .

$5 (x (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 3.2

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

$5 (x (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 3.3

Replace all occurrences of $u$ with $x - 8$ .

$5 (x (\frac{1}{3} {(x - 8)}^{\frac{1}{3} - 1} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 4

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

$5 (x (\frac{1}{3} {(x - 8)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 5

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

$5 (x (\frac{1}{3} {(x - 8)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 6

Combine the numerators over the common denominator.

$5 (x (\frac{1}{3} {(x - 8)}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

$5 (x (\frac{1}{3} {(x - 8)}^{\frac{1 - 3}{3}} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 7.2

Subtract $3$ from $1$ .

$5 (x (\frac{1}{3} {(x - 8)}^{\frac{- 2}{3}} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$5 (x (\frac{1}{3} {(x - 8)}^{- \frac{2}{3}} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 8.2

Combine $\frac{1}{3}$ and ${(x - 8)}^{- \frac{2}{3}}$ .

$5 (x (\frac{{(x - 8)}^{- \frac{2}{3}}}{3} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 8.3

Move ${(x - 8)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$5 (x (\frac{1}{3 {(x - 8)}^{\frac{2}{3}}} \frac{d}{d x} [x - 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 8.4

Combine $\frac{1}{3 {(x - 8)}^{\frac{2}{3}}}$ and $x$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} \frac{d}{d x} [x - 8] + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 9

By the Sum Rule, the derivative of $x - 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 10

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

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} (1 + \frac{d}{d x} [- 8]) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 11

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} (1 + 0) + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $0$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} \cdot 1 + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 12.2

Multiply $\frac{x}{3 {(x - 8)}^{\frac{2}{3}}}$ by $1$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} + {(x - 8)}^{\frac{1}{3}} \frac{d}{d x} [x])$

Step 13

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

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} + {(x - 8)}^{\frac{1}{3}} \cdot 1)$

Step 14

Multiply ${(x - 8)}^{\frac{1}{3}}$ by $1$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} + {(x - 8)}^{\frac{1}{3}})$

Step 15

To write ${(x - 8)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(x - 8)}^{\frac{2}{3}}}{3 {(x - 8)}^{\frac{2}{3}}}$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} + {(x - 8)}^{\frac{1}{3}} \cdot \frac{3 {(x - 8)}^{\frac{2}{3}}}{3 {(x - 8)}^{\frac{2}{3}}})$

Step 16

Combine ${(x - 8)}^{\frac{1}{3}}$ and $\frac{3 {(x - 8)}^{\frac{2}{3}}}{3 {(x - 8)}^{\frac{2}{3}}}$ .

$5 (\frac{x}{3 {(x - 8)}^{\frac{2}{3}}} + \frac{{(x - 8)}^{\frac{1}{3}} (3 {(x - 8)}^{\frac{2}{3}})}{3 {(x - 8)}^{\frac{2}{3}}})$

Step 17

Combine the numerators over the common denominator.

$5 \frac{x + {(x - 8)}^{\frac{1}{3}} (3 {(x - 8)}^{\frac{2}{3}})}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 18

Multiply ${(x - 8)}^{\frac{1}{3}}$ by ${(x - 8)}^{\frac{2}{3}}$ by adding the exponents.

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Step 18.1

Move ${(x - 8)}^{\frac{2}{3}}$ .

$5 \frac{x + {(x - 8)}^{\frac{2}{3}} {(x - 8)}^{\frac{1}{3}} \cdot 3}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 18.2

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

$5 \frac{x + {(x - 8)}^{\frac{2}{3} + \frac{1}{3}} \cdot 3}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 18.3

Combine the numerators over the common denominator.

$5 \frac{x + {(x - 8)}^{\frac{2 + 1}{3}} \cdot 3}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 18.4

Add $2$ and $1$ .

$5 \frac{x + {(x - 8)}^{\frac{3}{3}} \cdot 3}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 18.5

Divide $3$ by $3$ .

$5 \frac{x + {(x - 8)}^{1} \cdot 3}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 19

Simplify the expression.

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Step 19.1

Simplify ${(x - 8)}^{1} \cdot 3$ .

$5 \frac{x + (x - 8) \cdot 3}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 19.2

Move $3$ to the left of $x - 8$ .

$5 \frac{x + 3 \cdot (x - 8)}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 20

Combine $5$ and $\frac{x + 3 (x - 8)}{3 {(x - 8)}^{\frac{2}{3}}}$ .

$\frac{5 (x + 3 (x - 8))}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21

Simplify.

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Step 21.1

Apply the distributive property.

$\frac{5 (x + 3 x + 3 \cdot - 8)}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.2

Apply the distributive property.

$\frac{5 x + 5 (3 x) + 5 (3 \cdot - 8)}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.3

Simplify the numerator.

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Step 21.3.1

Simplify each term.

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Step 21.3.1.1

Multiply $3$ by $5$ .

$\frac{5 x + 15 x + 5 (3 \cdot - 8)}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.3.1.2

Multiply $5 (3 \cdot - 8)$ .

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Step 21.3.1.2.1

Multiply $3$ by $- 8$ .

$\frac{5 x + 15 x + 5 \cdot - 24}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.3.1.2.2

Multiply $5$ by $- 24$ .

$\frac{5 x + 15 x - 120}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.3.2

Add $5 x$ and $15 x$ .

$\frac{20 x - 120}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.4

Factor $20$ out of $20 x - 120$ .

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Step 21.4.1

Factor $20$ out of $20 x$ .

$\frac{20 (x) - 120}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.4.2

Factor $20$ out of $- 120$ .

$\frac{20 x + 20 \cdot - 6}{3 {(x - 8)}^{\frac{2}{3}}}$

Step 21.4.3

Factor $20$ out of $20 x + 20 \cdot - 6$ .

$\frac{20 (x - 6)}{3 {(x - 8)}^{\frac{2}{3}}}$