Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = {(8 x^{3} - 3 x)}^{\frac{1}{2}}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(8 x^{3} - 3 x)}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{(8 x^{3} - 3 x)}^{\frac{1}{2}}]$

Step 1.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with ${(8 x^{3} - 3 x)}^{\frac{1}{2}}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 8 x^{3} - 3 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $8 x^{3} - 3 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $8 x^{3} - 3 x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Move ${(8 x^{3} - 3 x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10

Multiply $\frac{1}{2 {(8 x^{3} - 3 x)}^{\frac{1}{2}}}$ by $\frac{1}{{(8 x^{3} - 3 x)}^{\frac{1}{2}}}$ .

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

Step 11

Multiply ${(8 x^{3} - 3 x)}^{\frac{1}{2}}$ by ${(8 x^{3} - 3 x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 11.1

Move ${(8 x^{3} - 3 x)}^{\frac{1}{2}}$ .

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

Step 11.2

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

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

Step 11.3

Combine the numerators over the common denominator.

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

Step 11.4

Add $1$ and $1$ .

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

Step 11.5

Divide $2$ by $2$ .

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

Step 12

Simplify $2 {(8 x^{3} - 3 x)}^{1}$ .

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Multiply $3$ by $8$ .

$\frac{1}{2 (8 x^{3} - 3 x)} (24 x^{2} + \frac{d}{d x} [- 3 x])$

Step 17

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

$\frac{1}{2 (8 x^{3} - 3 x)} (24 x^{2} - 3 \frac{d}{d x} [x])$

Step 18

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

$\frac{1}{2 (8 x^{3} - 3 x)} (24 x^{2} - 3 \cdot 1)$

Step 19

Multiply $- 3$ by $1$ .

$\frac{1}{2 (8 x^{3} - 3 x)} (24 x^{2} - 3)$

Step 20

Simplify.

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Step 20.1

Apply the distributive property.

$\frac{1}{2 (8 x^{3}) + 2 (- 3 x)} (24 x^{2} - 3)$

Step 20.2

Combine terms.

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Step 20.2.1

Multiply $8$ by $2$ .

$\frac{1}{16 x^{3} + 2 (- 3 x)} (24 x^{2} - 3)$

Step 20.2.2

Multiply $- 3$ by $2$ .

$\frac{1}{16 x^{3} - 6 x} (24 x^{2} - 3)$

Step 20.3

Reorder the factors of $\frac{1}{16 x^{3} - 6 x} (24 x^{2} - 3)$ .

$(24 x^{2} - 3) \frac{1}{16 x^{3} - 6 x}$

Step 20.4

Factor $2 x$ out of $16 x^{3} - 6 x$ .

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Step 20.4.1

Factor $2 x$ out of $16 x^{3}$ .

$(24 x^{2} - 3) \frac{1}{2 x (8 x^{2}) - 6 x}$

Step 20.4.2

Factor $2 x$ out of $- 6 x$ .

$(24 x^{2} - 3) \frac{1}{2 x (8 x^{2}) + 2 x (- 3)}$

Step 20.4.3

Factor $2 x$ out of $2 x (8 x^{2}) + 2 x (- 3)$ .

$(24 x^{2} - 3) \frac{1}{2 x (8 x^{2} - 3)}$

Step 20.5

Multiply $24 x^{2} - 3$ by $\frac{1}{2 x (8 x^{2} - 3)}$ .

$\frac{24 x^{2} - 3}{2 x (8 x^{2} - 3)}$

Step 20.6

Factor $3$ out of $24 x^{2} - 3$ .

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Step 20.6.1

Factor $3$ out of $24 x^{2}$ .

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

Step 20.6.2

Factor $3$ out of $- 3$ .

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

Step 20.6.3

Factor $3$ out of $3 (8 x^{2}) + 3 (- 1)$ .

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