Calculus Examples

$f (x) = 4 \sqrt[3]{x} - \frac{3}{4 x^{6}}$

Step 1

By the Sum Rule, the derivative of $4 \sqrt[3]{x} - \frac{3}{4 x^{6}}$ with respect to $x$ is $\frac{d}{d x} [4 \sqrt[3]{x}] + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$ .

$\frac{d}{d x} [4 \sqrt[3]{x}] + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2

Evaluate $\frac{d}{d x} [4 \sqrt[3]{x}]$ .

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Step 2.1

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

$\frac{d}{d x} [4 x^{\frac{1}{3}}] + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{1}{3}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

$4 \frac{d}{d x} [x^{\frac{1}{3}}] + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.3

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

$4 (\frac{1}{3} x^{\frac{1}{3} - 1}) + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.4

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

$4 (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.5

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

$4 (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.6

Combine the numerators over the common denominator.

$4 (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}) + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

$4 (\frac{1}{3} x^{\frac{1 - 3}{3}}) + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.7.2

Subtract $3$ from $1$ .

$4 (\frac{1}{3} x^{\frac{- 2}{3}}) + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.8

Move the negative in front of the fraction.

$4 (\frac{1}{3} x^{- \frac{2}{3}}) + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.9

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

$4 \frac{x^{- \frac{2}{3}}}{3} + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.10

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

$\frac{4 x^{- \frac{2}{3}}}{3} + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 2.11

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

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{d}{d x} [- \frac{3}{4 x^{6}}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{3}{4 x^{6}}]$ .

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Step 3.1

Since $- \frac{3}{4}$ is constant with respect to $x$ , the derivative of $- \frac{3}{4 x^{6}}$ with respect to $x$ is $- \frac{3}{4} \frac{d}{d x} [\frac{1}{x^{6}}]$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} \frac{d}{d x} [\frac{1}{x^{6}}]$

Step 3.2

Rewrite $\frac{1}{x^{6}}$ as ${(x^{6})}^{- 1}$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} \frac{d}{d x} [{(x^{6})}^{- 1}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{6}$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x^{6}$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{6}])$

Step 3.3.2

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

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- u^{- 2} \frac{d}{d x} [x^{6}])$

Step 3.3.3

Replace all occurrences of $u$ with $x^{6}$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- {(x^{6})}^{- 2} \frac{d}{d x} [x^{6}])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- {(x^{6})}^{- 2} (6 x^{5}))$

Step 3.5

Multiply the exponents in ${(x^{6})}^{- 2}$ .

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Step 3.5.1

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

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- x^{6 \cdot - 2} (6 x^{5}))$

Step 3.5.2

Multiply $6$ by $- 2$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- x^{- 12} (6 x^{5}))$

Step 3.6

Multiply $6$ by $- 1$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- 6 x^{- 12} x^{5})$

Step 3.7

Multiply $x^{- 12}$ by $x^{5}$ by adding the exponents.

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Step 3.7.1

Move $x^{5}$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- 6 (x^{5} x^{- 12}))$

Step 3.7.2

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

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- 6 x^{5 - 12})$

Step 3.7.3

Subtract $12$ from $5$ .

$\frac{4}{3 x^{\frac{2}{3}}} - \frac{3}{4} (- 6 x^{- 7})$

Step 3.8

Multiply $- 6$ by $- 1$ .

$\frac{4}{3 x^{\frac{2}{3}}} + 6 (\frac{3}{4}) x^{- 7}$

Step 3.9

Combine $6$ and $\frac{3}{4}$ .

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{6 \cdot 3}{4} x^{- 7}$

Step 3.10

Multiply $6$ by $3$ .

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{18}{4} x^{- 7}$

Step 3.11

Combine $\frac{18}{4}$ and $x^{- 7}$ .

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{18 x^{- 7}}{4}$

Step 3.12

Move $x^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{18}{4 x^{7}}$

Step 3.13

Cancel the common factor of $18$ and $4$ .

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Step 3.13.1

Factor $2$ out of $18$ .

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{2 (9)}{4 x^{7}}$

Step 3.13.2

Cancel the common factors.

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Step 3.13.2.1

Factor $2$ out of $4 x^{7}$ .

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{2 (9)}{2 (2 x^{7})}$

Step 3.13.2.2

Cancel the common factor.

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{2 \cdot 9}{2 (2 x^{7})}$

Step 3.13.2.3

Rewrite the expression.

$\frac{4}{3 x^{\frac{2}{3}}} + \frac{9}{2 x^{7}}$

Step 4

Reorder terms.

$\frac{9}{2 x^{7}} + \frac{4}{3 x^{\frac{2}{3}}}$