Calculus Examples

$\frac{7}{\sqrt[3]{x^{2}}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 7$ and $g (x) = \sqrt[3]{x^{2}}$ .

$\frac{\sqrt[3]{x^{2}} \frac{d}{d x} [7] - 1 \cdot 7 \frac{d}{d x} [\sqrt[3]{x^{2}}]}{{\sqrt[3]{x^{2}}}^{2}}$

Step 2

Differentiate.

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 \frac{d}{d x} [\sqrt[3]{x^{2}}]}{{\sqrt[3]{x^{2}}}^{2}}$

Step 2.2

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

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 \frac{d}{d x} [x^{\frac{2}{3}}]}{{\sqrt[3]{x^{2}}}^{2}}$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} x^{\frac{2}{3} - 1})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 3

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

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 4

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

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $3$ .

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} x^{\frac{2 - 3}{3}})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 6.2

Subtract $3$ from $2$ .

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} x^{\frac{- 1}{3}})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 7

Move the negative in front of the fraction.

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} x^{- \frac{1}{3}})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 8

Simplify.

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Step 8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 (\frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}})}{{\sqrt[3]{x^{2}}}^{2}}$

Step 8.2

Multiply $\frac{2}{3}$ by $\frac{1}{x^{\frac{1}{3}}}$ .

$\frac{\sqrt[3]{x^{2}} \cdot 0 - 1 \cdot 7 \frac{2}{3 x^{\frac{1}{3}}}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Simplify each term.

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Step 9.1.1.1

Multiply $\sqrt[3]{x^{2}}$ by $0$ .

$\frac{0 - 1 \cdot 7 \frac{2}{3 x^{\frac{1}{3}}}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 9.1.1.2

Multiply $- 1$ by $7$ .

$\frac{0 - 7 \frac{2}{3 x^{\frac{1}{3}}}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 9.1.1.3

Multiply $- 7 \frac{2}{3 x^{\frac{1}{3}}}$ .

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Step 9.1.1.3.1

Combine $- 7$ and $\frac{2}{3 x^{\frac{1}{3}}}$ .

$\frac{0 + \frac{- 7 \cdot 2}{3 x^{\frac{1}{3}}}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 9.1.1.3.2

Multiply $- 7$ by $2$ .

$\frac{0 + \frac{- 14}{3 x^{\frac{1}{3}}}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 9.1.1.4

Move the negative in front of the fraction.

$\frac{0 - \frac{14}{3 x^{\frac{1}{3}}}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 9.1.2

Subtract $\frac{14}{3 x^{\frac{1}{3}}}$ from $0$ .

$\frac{- \frac{14}{3 x^{\frac{1}{3}}}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 9.2

Combine terms.

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Step 9.2.1

Rewrite ${\sqrt[3]{x^{2}}}^{2}$ as $\sqrt[3]{{(x^{2})}^{2}}$ .

$\frac{- \frac{14}{3 x^{\frac{1}{3}}}}{\sqrt[3]{{(x^{2})}^{2}}}$

Step 9.2.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 9.2.2.1

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

$\frac{- \frac{14}{3 x^{\frac{1}{3}}}}{\sqrt[3]{x^{2 \cdot 2}}}$

Step 9.2.2.2

Multiply $2$ by $2$ .

$\frac{- \frac{14}{3 x^{\frac{1}{3}}}}{\sqrt[3]{x^{4}}}$

Step 9.2.3

Rewrite $\frac{- \frac{14}{3 x^{\frac{1}{3}}}}{\sqrt[3]{x^{4}}}$ as a product.

$- \frac{14}{3 x^{\frac{1}{3}}} \cdot \frac{1}{\sqrt[3]{x^{4}}}$

Step 9.2.4

Multiply $\frac{1}{\sqrt[3]{x^{4}}}$ by $\frac{14}{3 x^{\frac{1}{3}}}$ .

$- \frac{14}{\sqrt[3]{x^{4}} (3 x^{\frac{1}{3}})}$

Step 9.2.5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$ .

$- \frac{14}{3 (x^{\frac{4}{3}} x^{\frac{1}{3}})}$

Step 9.2.6

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

$- \frac{14}{3 x^{\frac{4}{3} + \frac{1}{3}}}$

Step 9.2.7

Combine the numerators over the common denominator.

$- \frac{14}{3 x^{\frac{4 + 1}{3}}}$

Step 9.2.8

Add $4$ and $1$ .

$- \frac{14}{3 x^{\frac{5}{3}}}$