Calculus Examples

$y = \frac{3}{x} + \frac{2}{\sqrt[3]{x^{2}}}$

Step 1

By the Sum Rule, the derivative of $\frac{3}{x} + \frac{2}{\sqrt[3]{x^{2}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{x}] + \frac{d}{d x} [\frac{2}{\sqrt[3]{x^{2}}}]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\frac{3}{x}]$ .

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Step 2.1

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

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

Step 2.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.3

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

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

Step 2.4

Multiply $- 1$ by $3$ .

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

Step 3

Evaluate $\frac{d}{d x} [\frac{2}{\sqrt[3]{x^{2}}}]$ .

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Step 3.1

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

$- 3 x^{- 2} + \frac{d}{d x} [\frac{2}{x^{\frac{2}{3}}}]$

Step 3.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{\frac{2}{3}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

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

Step 3.3

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

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

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{2}{3}}$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}}$ .

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

Step 3.4.2

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

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

Step 3.4.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

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

Step 3.5

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

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 3.6.2.1

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

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

Step 3.6.2.2

Multiply $2$ by $- 2$ .

$- 3 x^{- 2} + 2 (- x^{\frac{- 4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 3.6.3

Move the negative in front of the fraction.

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 3.7

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

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}))$

Step 3.8

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

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 3.9

Combine the numerators over the common denominator.

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}))$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $3$ .

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 3}{3}}))$

Step 3.10.2

Subtract $3$ from $2$ .

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{- 1}{3}}))$

Step 3.11

Move the negative in front of the fraction.

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} (\frac{2}{3} x^{- \frac{1}{3}}))$

Step 3.12

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

$- 3 x^{- 2} + 2 (- x^{- \frac{4}{3}} \frac{2 x^{- \frac{1}{3}}}{3})$

Step 3.13

Combine $\frac{2 x^{- \frac{1}{3}}}{3}$ and $x^{- \frac{4}{3}}$ .

$- 3 x^{- 2} + 2 (- \frac{2 x^{- \frac{1}{3}} x^{- \frac{4}{3}}}{3})$

Step 3.14

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 3.14.1

Move $x^{- \frac{4}{3}}$ .

$- 3 x^{- 2} + 2 (- \frac{2 (x^{- \frac{4}{3}} x^{- \frac{1}{3}})}{3})$

Step 3.14.2

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

$- 3 x^{- 2} + 2 (- \frac{2 x^{- \frac{4}{3} - \frac{1}{3}}}{3})$

Step 3.14.3

Combine the numerators over the common denominator.

$- 3 x^{- 2} + 2 (- \frac{2 x^{\frac{- 4 - 1}{3}}}{3})$

Step 3.14.4

Subtract $1$ from $- 4$ .

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

Step 3.14.5

Move the negative in front of the fraction.

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

Step 3.15

Move $x^{- \frac{5}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.16

Multiply $- 1$ by $2$ .

$- 3 x^{- 2} - 2 \frac{2}{3 x^{\frac{5}{3}}}$

Step 3.17

Combine $- 2$ and $\frac{2}{3 x^{\frac{5}{3}}}$ .

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

Step 3.18

Multiply $- 2$ by $2$ .

$- 3 x^{- 2} + \frac{- 4}{3 x^{\frac{5}{3}}}$

Step 3.19

Move the negative in front of the fraction.

$- 3 x^{- 2} - \frac{4}{3 x^{\frac{5}{3}}}$

Step 4

Simplify.

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Step 4.1

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

$- 3 \frac{1}{x^{2}} - \frac{4}{3 x^{\frac{5}{3}}}$

Step 4.2

Combine terms.

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Step 4.2.1

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

$\frac{- 3}{x^{2}} - \frac{4}{3 x^{\frac{5}{3}}}$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{3}{x^{2}} - \frac{4}{3 x^{\frac{5}{3}}}$