Calculus Examples

$y = \frac{1}{\sqrt[3]{{(8 - x^{3})}^{8}}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{\sqrt[3]{{(8 - x^{3})}^{8}}}]$ as $\frac{d}{d x} [\frac{1}{{(8 - x^{3})}^{2} \sqrt[3]{{(8 - x^{3})}^{2}}}]$ .

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Step 1.1

Rewrite ${(8 - x^{3})}^{8}$ as ${({(8 - x^{3})}^{2})}^{3} {(8 - x^{3})}^{2}$ .

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Step 1.1.1

Factor out ${(8 - x^{3})}^{6}$ .

$\frac{d}{d x} [\frac{1}{\sqrt[3]{{(8 - x^{3})}^{6} {(8 - x^{3})}^{2}}}]$

Step 1.1.2

Rewrite ${(8 - x^{3})}^{6}$ as ${({(8 - x^{3})}^{2})}^{3}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(8 - x^{3})}^{2}}$ as ${(8 - x^{3})}^{\frac{2}{3}}$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $3$ .

$\frac{d}{d x} [\frac{1}{{(8 - x^{3})}^{\frac{6 + 2}{3}}}]$

Step 3.5.2

Add $6$ and $2$ .

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

Step 4

Apply basic rules of exponents.

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Step 4.1

Rewrite $\frac{1}{{(8 - x^{3})}^{\frac{8}{3}}}$ as ${({(8 - x^{3})}^{\frac{8}{3}})}^{- 1}$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $\frac{8}{3} \cdot - 1$ .

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Step 4.2.2.1

Combine $\frac{8}{3}$ and $- 1$ .

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

Step 4.2.2.2

Multiply $8$ by $- 1$ .

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

Step 4.2.3

Move the negative in front of the fraction.

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

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{8}{3}}$ and $g (x) = 8 - x^{3}$ .

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Step 5.1

To apply the Chain Rule, set $u$ as $8 - x^{3}$ .

$\frac{d}{d u} [u^{- \frac{8}{3}}] \frac{d}{d x} [8 - x^{3}]$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{8}{3}$ .

$- \frac{8}{3} u^{- \frac{8}{3} - 1} \frac{d}{d x} [8 - x^{3}]$

Step 5.3

Replace all occurrences of $u$ with $8 - x^{3}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $3$ .

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

Step 9.2

Subtract $3$ from $- 8$ .

$- \frac{8}{3} {(8 - x^{3})}^{\frac{- 11}{3}} \frac{d}{d x} [8 - x^{3}]$

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

$- \frac{8}{3} {(8 - x^{3})}^{- \frac{11}{3}} \frac{d}{d x} [8 - x^{3}]$

Step 10.2

Combine ${(8 - x^{3})}^{- \frac{11}{3}}$ and $\frac{8}{3}$ .

$- \frac{{(8 - x^{3})}^{- \frac{11}{3}} \cdot 8}{3} \frac{d}{d x} [8 - x^{3}]$

Step 10.3

Simplify the expression.

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Step 10.3.1

Move $8$ to the left of ${(8 - x^{3})}^{- \frac{11}{3}}$ .

$- \frac{8 \cdot {(8 - x^{3})}^{- \frac{11}{3}}}{3} \frac{d}{d x} [8 - x^{3}]$

Step 10.3.2

Move ${(8 - x^{3})}^{- \frac{11}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} \frac{d}{d x} [8 - x^{3}]$

Step 11

By the Sum Rule, the derivative of $8 - x^{3}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- x^{3}]$ .

$- \frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} (\frac{d}{d x} [8] + \frac{d}{d x} [- x^{3}])$

Step 12

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

$- \frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} (0 + \frac{d}{d x} [- x^{3}])$

Step 13

Add $0$ and $\frac{d}{d x} [- x^{3}]$ .

$- \frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} \frac{d}{d x} [- x^{3}]$

Step 14

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

$- \frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} (- \frac{d}{d x} [x^{3}])$

Step 15

Multiply.

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Step 15.1

Multiply $- 1$ by $- 1$ .

$1 \frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} \frac{d}{d x} [x^{3}]$

Step 15.2

Multiply $\frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}}$ by $1$ .

$\frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} \frac{d}{d x} [x^{3}]$

Step 16

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

$\frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}} (3 x^{2})$

Step 17

Simplify terms.

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Step 17.1

Combine $3$ and $\frac{8}{3 {(8 - x^{3})}^{\frac{11}{3}}}$ .

$\frac{3 \cdot 8}{3 {(8 - x^{3})}^{\frac{11}{3}}} x^{2}$

Step 17.2

Multiply $3$ by $8$ .

$\frac{24}{3 {(8 - x^{3})}^{\frac{11}{3}}} x^{2}$

Step 17.3

Combine $\frac{24}{3 {(8 - x^{3})}^{\frac{11}{3}}}$ and $x^{2}$ .

$\frac{24 x^{2}}{3 {(8 - x^{3})}^{\frac{11}{3}}}$

Step 17.4

Factor $3$ out of $24 x^{2}$ .

$\frac{3 (8 x^{2})}{3 {(8 - x^{3})}^{\frac{11}{3}}}$

Step 18

Cancel the common factors.

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Step 18.1

Factor $3$ out of $3 {(8 - x^{3})}^{\frac{11}{3}}$ .

$\frac{3 (8 x^{2})}{3 ({(8 - x^{3})}^{\frac{11}{3}})}$

Step 18.2

Cancel the common factor.

$\frac{3 (8 x^{2})}{3 {(8 - x^{3})}^{\frac{11}{3}}}$

Step 18.3

Rewrite the expression.

$\frac{8 x^{2}}{{(8 - x^{3})}^{\frac{11}{3}}}$

Step 19

Reorder terms.

$\frac{8 x^{2}}{{(- x^{3} + 8)}^{\frac{11}{3}}}$