Calculus Examples

$f (x) = 8 \sqrt{x^{3}} - \frac{8}{\sqrt[4]{x}}$

Step 1

By the Sum Rule, the derivative of $8 \sqrt{x^{3}} - \frac{8}{\sqrt[4]{x}}$ with respect to $x$ is $\frac{d}{d x} [8 \sqrt{x^{3}}] + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$ .

$\frac{d}{d x} [8 \sqrt{x^{3}}] + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2

Evaluate $\frac{d}{d x} [8 \sqrt{x^{3}}]$ .

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Step 2.1

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

$\frac{d}{d x} [8 x^{\frac{3}{2}}] + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{\frac{3}{2}}$ with respect to $x$ is $8 \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$8 \frac{d}{d x} [x^{\frac{3}{2}}] + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.3

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

$8 (\frac{3}{2} x^{\frac{3}{2} - 1}) + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.5

Combine $- 1$ and $\frac{2}{2}$ .

$8 (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.6

Combine the numerators over the common denominator.

$8 (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$8 (\frac{3}{2} x^{\frac{3 - 2}{2}}) + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.7.2

Subtract $2$ from $3$ .

$8 (\frac{3}{2} x^{\frac{1}{2}}) + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.8

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

$8 \frac{3 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.9

Combine $8$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$\frac{8 (3 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.10

Multiply $3$ by $8$ .

$\frac{24 x^{\frac{1}{2}}}{2} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.11

Factor $2$ out of $24 x^{\frac{1}{2}}$ .

$\frac{2 (12 x^{\frac{1}{2}})}{2} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $2$ out of $2$ .

$\frac{2 (12 x^{\frac{1}{2}})}{2 (1)} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.12.2

Cancel the common factor.

$\frac{2 (12 x^{\frac{1}{2}})}{2 \cdot 1} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.12.3

Rewrite the expression.

$\frac{12 x^{\frac{1}{2}}}{1} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 2.12.4

Divide $12 x^{\frac{1}{2}}$ by $1$ .

$12 x^{\frac{1}{2}} + \frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{8}{\sqrt[4]{x}}]$ .

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$12 x^{\frac{1}{2}} + \frac{d}{d x} [- \frac{8}{x^{\frac{1}{4}}}]$

Step 3.2

Since $- 8$ is constant with respect to $x$ , the derivative of $- \frac{8}{x^{\frac{1}{4}}}$ with respect to $x$ is $- 8 \frac{d}{d x} [\frac{1}{x^{\frac{1}{4}}}]$ .

$12 x^{\frac{1}{2}} - 8 \frac{d}{d x} [\frac{1}{x^{\frac{1}{4}}}]$

Step 3.3

Rewrite $\frac{1}{x^{\frac{1}{4}}}$ as ${(x^{\frac{1}{4}})}^{- 1}$ .

$12 x^{\frac{1}{2}} - 8 \frac{d}{d x} [{(x^{\frac{1}{4}})}^{- 1}]$

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 $g (x) = x^{\frac{1}{4}}$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{4}}$ .

$12 x^{\frac{1}{2}} - 8 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{1}{4}}])$

Step 3.4.2

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

$12 x^{\frac{1}{2}} - 8 (- u^{- 2} \frac{d}{d x} [x^{\frac{1}{4}}])$

Step 3.4.3

Replace all occurrences of $u$ with $x^{\frac{1}{4}}$ .

$12 x^{\frac{1}{2}} - 8 (- {(x^{\frac{1}{4}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{4}}])$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{4}$ .

$12 x^{\frac{1}{2}} - 8 (- {(x^{\frac{1}{4}})}^{- 2} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.6

Multiply the exponents in ${(x^{\frac{1}{4}})}^{- 2}$ .

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Step 3.6.1

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

$12 x^{\frac{1}{2}} - 8 (- x^{\frac{1}{4} \cdot - 2} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.6.2

Cancel the common factor of $2$ .

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Step 3.6.2.1

Factor $2$ out of $4$ .

$12 x^{\frac{1}{2}} - 8 (- x^{\frac{1}{2 (2)} \cdot - 2} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.6.2.2

Factor $2$ out of $- 2$ .

$12 x^{\frac{1}{2}} - 8 (- x^{\frac{1}{2 \cdot 2} \cdot (2 \cdot - 1)} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.6.2.3

Cancel the common factor.

$12 x^{\frac{1}{2}} - 8 (- x^{\frac{1}{2 \cdot 2} \cdot (2 \cdot - 1)} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.6.2.4

Rewrite the expression.

$12 x^{\frac{1}{2}} - 8 (- x^{\frac{1}{2} \cdot - 1} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.6.3

Combine $\frac{1}{2}$ and $- 1$ .

$12 x^{\frac{1}{2}} - 8 (- x^{\frac{- 1}{2}} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.6.4

Move the negative in front of the fraction.

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} (\frac{1}{4} x^{\frac{1}{4} - 1}))$

Step 3.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} (\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}}))$

Step 3.8

Combine $- 1$ and $\frac{4}{4}$ .

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} (\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}))$

Step 3.9

Combine the numerators over the common denominator.

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} (\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}}))$

Step 3.10

Simplify the numerator.

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Step 3.10.1

Multiply $- 1$ by $4$ .

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} (\frac{1}{4} x^{\frac{1 - 4}{4}}))$

Step 3.10.2

Subtract $4$ from $1$ .

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} (\frac{1}{4} x^{\frac{- 3}{4}}))$

Step 3.11

Move the negative in front of the fraction.

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} (\frac{1}{4} x^{- \frac{3}{4}}))$

Step 3.12

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

$12 x^{\frac{1}{2}} - 8 (- x^{- \frac{1}{2}} \frac{x^{- \frac{3}{4}}}{4})$

Step 3.13

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

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{- \frac{3}{4}} x^{- \frac{1}{2}}}{4})$

Step 3.14

Multiply $x^{- \frac{3}{4}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.14.1

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

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{- \frac{3}{4} - \frac{1}{2}}}{4})$

Step 3.14.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{- \frac{3}{4} - \frac{1}{2} \cdot \frac{2}{2}}}{4})$

Step 3.14.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.14.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{- \frac{3}{4} - \frac{2}{2 \cdot 2}}}{4})$

Step 3.14.3.2

Multiply $2$ by $2$ .

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{- \frac{3}{4} - \frac{2}{4}}}{4})$

Step 3.14.4

Combine the numerators over the common denominator.

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{\frac{- 3 - 2}{4}}}{4})$

Step 3.14.5

Subtract $2$ from $- 3$ .

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{\frac{- 5}{4}}}{4})$

Step 3.14.6

Move the negative in front of the fraction.

$12 x^{\frac{1}{2}} - 8 (- \frac{x^{- \frac{5}{4}}}{4})$

Step 3.15

Move $x^{- \frac{5}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 x^{\frac{1}{2}} - 8 (- \frac{1}{4 x^{\frac{5}{4}}})$

Step 3.16

Multiply $- 1$ by $- 8$ .

$12 x^{\frac{1}{2}} + 8 \frac{1}{4 x^{\frac{5}{4}}}$

Step 3.17

Combine $8$ and $\frac{1}{4 x^{\frac{5}{4}}}$ .

$12 x^{\frac{1}{2}} + \frac{8}{4 x^{\frac{5}{4}}}$

Step 3.18

Factor $4$ out of $8$ .

$12 x^{\frac{1}{2}} + \frac{4 \cdot 2}{4 x^{\frac{5}{4}}}$

Step 3.19

Cancel the common factors.

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Step 3.19.1

Factor $4$ out of $4 x^{\frac{5}{4}}$ .

$12 x^{\frac{1}{2}} + \frac{4 \cdot 2}{4 (x^{\frac{5}{4}})}$

Step 3.19.2

Cancel the common factor.

$12 x^{\frac{1}{2}} + \frac{4 \cdot 2}{4 x^{\frac{5}{4}}}$

Step 3.19.3

Rewrite the expression.

$12 x^{\frac{1}{2}} + \frac{2}{x^{\frac{5}{4}}}$