Calculus Examples

$\frac{14}{\sqrt[3]{14 x + 1}}$

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 $f (x) = 14$ and $g (x) = \sqrt[3]{14 x + 1}$ .

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

Step 2

Differentiate using the Constant Rule.

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Step 2.1

Since $14$ is constant with respect to $x$ , the derivative of $14$ with respect to $x$ is $0$ .

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

Step 2.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{14 x + 1}$ as ${(14 x + 1)}^{\frac{1}{3}}$ .

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

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 $g (x) = 14 x + 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $14 x + 1$ .

$\frac{\sqrt[3]{14 x + 1} \cdot 0 - 1 \cdot 14 (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d x} [14 x + 1])}{{\sqrt[3]{14 x + 1}}^{2}}$

Step 3.2

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

$\frac{\sqrt[3]{14 x + 1} \cdot 0 - 1 \cdot 14 (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d x} [14 x + 1])}{{\sqrt[3]{14 x + 1}}^{2}}$

Step 3.3

Replace all occurrences of $u$ with $14 x + 1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $3$ .

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

Step 7.2

Subtract $3$ from $1$ .

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

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

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

Step 8.2

Combine $\frac{1}{3}$ and ${(14 x + 1)}^{- \frac{2}{3}}$ .

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

Step 8.3

Move ${(14 x + 1)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9

By the Sum Rule, the derivative of $14 x + 1$ with respect to $x$ is $\frac{d}{d x} [14 x] + \frac{d}{d x} [1]$ .

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

Step 10

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

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

Step 11

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

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

Step 12

Multiply $14$ by $1$ .

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

Step 13

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

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

Step 14

Combine fractions.

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Step 14.1

Add $14$ and $0$ .

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

Step 14.2

Combine $\frac{1}{3 {(14 x + 1)}^{\frac{2}{3}}}$ and $14$ .

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

Step 15

Simplify.

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Step 15.1

Simplify the numerator.

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Step 15.1.1

Simplify each term.

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Step 15.1.1.1

Multiply $\sqrt[3]{14 x + 1}$ by $0$ .

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

Step 15.1.1.2

Multiply $- 1$ by $14$ .

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

Step 15.1.1.3

Multiply $- 14 \frac{14}{3 {(14 x + 1)}^{\frac{2}{3}}}$ .

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Step 15.1.1.3.1

Combine $- 14$ and $\frac{14}{3 {(14 x + 1)}^{\frac{2}{3}}}$ .

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

Step 15.1.1.3.2

Multiply $- 14$ by $14$ .

$\frac{0 + \frac{- 196}{3 {(14 x + 1)}^{\frac{2}{3}}}}{{\sqrt[3]{14 x + 1}}^{2}}$

Step 15.1.1.4

Move the negative in front of the fraction.

$\frac{0 - \frac{196}{3 {(14 x + 1)}^{\frac{2}{3}}}}{{\sqrt[3]{14 x + 1}}^{2}}$

Step 15.1.2

Subtract $\frac{196}{3 {(14 x + 1)}^{\frac{2}{3}}}$ from $0$ .

$\frac{- \frac{196}{3 {(14 x + 1)}^{\frac{2}{3}}}}{{\sqrt[3]{14 x + 1}}^{2}}$

Step 15.2

Combine terms.

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Step 15.2.1

Rewrite ${\sqrt[3]{14 x + 1}}^{2}$ as $\sqrt[3]{{(14 x + 1)}^{2}}$ .

$\frac{- \frac{196}{3 {(14 x + 1)}^{\frac{2}{3}}}}{\sqrt[3]{{(14 x + 1)}^{2}}}$

Step 15.2.2

Rewrite $\frac{- \frac{196}{3 {(14 x + 1)}^{\frac{2}{3}}}}{\sqrt[3]{{(14 x + 1)}^{2}}}$ as a product.

$- \frac{196}{3 {(14 x + 1)}^{\frac{2}{3}}} \cdot \frac{1}{\sqrt[3]{{(14 x + 1)}^{2}}}$

Step 15.2.3

Multiply $\frac{1}{\sqrt[3]{{(14 x + 1)}^{2}}}$ by $\frac{196}{3 {(14 x + 1)}^{\frac{2}{3}}}$ .

$- \frac{196}{\sqrt[3]{{(14 x + 1)}^{2}} (3 {(14 x + 1)}^{\frac{2}{3}})}$

Step 15.2.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(14 x + 1)}^{2}}$ as ${(14 x + 1)}^{\frac{2}{3}}$ .

$- \frac{196}{3 ({(14 x + 1)}^{\frac{2}{3}} {(14 x + 1)}^{\frac{2}{3}})}$

Step 15.2.5

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

$- \frac{196}{3 {(14 x + 1)}^{\frac{2}{3} + \frac{2}{3}}}$

Step 15.2.6

Combine the numerators over the common denominator.

$- \frac{196}{3 {(14 x + 1)}^{\frac{2 + 2}{3}}}$

Step 15.2.7

Add $2$ and $2$ .

$- \frac{196}{3 {(14 x + 1)}^{\frac{4}{3}}}$