Calculus Examples

$g (x) = {(33 x + 31)}^{\frac{4}{3}} \ln (x)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = {(33 x + 31)}^{\frac{4}{3}}$ and $g (x) = \ln (x)$ .

${(33 x + 31)}^{\frac{4}{3}} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [{(33 x + 31)}^{\frac{4}{3}}]$

Step 2

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

${(33 x + 31)}^{\frac{4}{3}} \frac{1}{x} + \ln (x) \frac{d}{d x} [{(33 x + 31)}^{\frac{4}{3}}]$

Step 3

Combine ${(33 x + 31)}^{\frac{4}{3}}$ and $\frac{1}{x}$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) \frac{d}{d x} [{(33 x + 31)}^{\frac{4}{3}}]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{4}{3}}$ and $g (x) = 33 x + 31$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $33 x + 31$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{d}{d u} [u^{\frac{4}{3}}] \frac{d}{d x} [33 x + 31])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{3}$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4}{3} u^{\frac{4}{3} - 1} \frac{d}{d x} [33 x + 31])$

Step 4.3

Replace all occurrences of $u$ with $33 x + 31$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4}{3} {(33 x + 31)}^{\frac{4}{3} - 1} \frac{d}{d x} [33 x + 31])$

Step 5

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

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4}{3} {(33 x + 31)}^{\frac{4}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [33 x + 31])$

Step 6

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

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4}{3} {(33 x + 31)}^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [33 x + 31])$

Step 7

Combine the numerators over the common denominator.

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4}{3} {(33 x + 31)}^{\frac{4 - 1 \cdot 3}{3}} \frac{d}{d x} [33 x + 31])$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $3$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4}{3} {(33 x + 31)}^{\frac{4 - 3}{3}} \frac{d}{d x} [33 x + 31])$

Step 8.2

Subtract $3$ from $4$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4}{3} {(33 x + 31)}^{\frac{1}{3}} \frac{d}{d x} [33 x + 31])$

Step 9

Combine fractions.

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Step 9.1

Combine $\frac{4}{3}$ and ${(33 x + 31)}^{\frac{1}{3}}$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \ln (x) (\frac{4 {(33 x + 31)}^{\frac{1}{3}}}{3} \frac{d}{d x} [33 x + 31])$

Step 9.2

Combine $\frac{4 {(33 x + 31)}^{\frac{1}{3}}}{3}$ and $\ln (x)$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3} \frac{d}{d x} [33 x + 31]$

Step 10

By the Sum Rule, the derivative of $33 x + 31$ with respect to $x$ is $\frac{d}{d x} [33 x] + \frac{d}{d x} [31]$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3} (\frac{d}{d x} [33 x] + \frac{d}{d x} [31])$

Step 11

Since $33$ is constant with respect to $x$ , the derivative of $33 x$ with respect to $x$ is $33 \frac{d}{d x} [x]$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3} (33 \frac{d}{d x} [x] + \frac{d}{d x} [31])$

Step 12

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

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3} (33 \cdot 1 + \frac{d}{d x} [31])$

Step 13

Multiply $33$ by $1$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3} (33 + \frac{d}{d x} [31])$

Step 14

Since $31$ is constant with respect to $x$ , the derivative of $31$ with respect to $x$ is $0$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3} (33 + 0)$

Step 15

Simplify terms.

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Step 15.1

Add $33$ and $0$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3} \cdot 33$

Step 15.2

Combine $\frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3}$ and $33$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{4 {(33 x + 31)}^{\frac{1}{3}} \ln (x) \cdot 33}{3}$

Step 15.3

Multiply $33$ by $4$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{132 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{3}$

Step 15.4

Factor $3$ out of $132 {(33 x + 31)}^{\frac{1}{3}} \ln (x)$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{3 (44 {(33 x + 31)}^{\frac{1}{3}} \ln (x))}{3}$

Step 16

Cancel the common factors.

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Step 16.1

Factor $3$ out of $3$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{3 (44 {(33 x + 31)}^{\frac{1}{3}} \ln (x))}{3 (1)}$

Step 16.2

Cancel the common factor.

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{3 (44 {(33 x + 31)}^{\frac{1}{3}} \ln (x))}{3 \cdot 1}$

Step 16.3

Rewrite the expression.

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{44 {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{1}$

Step 16.4

Divide $44 {(33 x + 31)}^{\frac{1}{3}} \ln (x)$ by $1$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + 44 {(33 x + 31)}^{\frac{1}{3}} \ln (x)$

Step 17

To write $44 {(33 x + 31)}^{\frac{1}{3}} \ln (x)$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}}}{x} + \frac{44 {(33 x + 31)}^{\frac{1}{3}} \ln (x) x}{x}$

Step 18

Combine the numerators over the common denominator.

$\frac{{(33 x + 31)}^{\frac{4}{3}} + 44 {(33 x + 31)}^{\frac{1}{3}} \ln (x) x}{x}$

Step 19

Simplify the numerator.

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Step 19.1

Factor ${(33 x + 31)}^{\frac{1}{3}}$ out of ${(33 x + 31)}^{\frac{4}{3}} + 44 {(33 x + 31)}^{\frac{1}{3}} \ln (x) x$ .

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Step 19.1.1

Reorder the expression.

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Step 19.1.1.1

Move $\ln (x)$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}} + 44 {(33 x + 31)}^{\frac{1}{3}} x \ln (x)}{x}$

Step 19.1.1.2

Move ${(33 x + 31)}^{\frac{1}{3}}$ .

$\frac{{(33 x + 31)}^{\frac{4}{3}} + 44 x {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{x}$

Step 19.1.2

Factor ${(33 x + 31)}^{\frac{1}{3}}$ out of ${(33 x + 31)}^{\frac{4}{3}}$ .

$\frac{{(33 x + 31)}^{\frac{1}{3}} {(33 x + 31)}^{\frac{3}{3}} + 44 x {(33 x + 31)}^{\frac{1}{3}} \ln (x)}{x}$

Step 19.1.3

Factor ${(33 x + 31)}^{\frac{1}{3}}$ out of $44 x {(33 x + 31)}^{\frac{1}{3}} \ln (x)$ .

$\frac{{(33 x + 31)}^{\frac{1}{3}} {(33 x + 31)}^{\frac{3}{3}} + {(33 x + 31)}^{\frac{1}{3}} (44 x \ln (x))}{x}$

Step 19.1.4

Factor ${(33 x + 31)}^{\frac{1}{3}}$ out of ${(33 x + 31)}^{\frac{1}{3}} {(33 x + 31)}^{\frac{3}{3}} + {(33 x + 31)}^{\frac{1}{3}} (44 x \ln (x))$ .

$\frac{{(33 x + 31)}^{\frac{1}{3}} ({(33 x + 31)}^{\frac{3}{3}} + 44 x \ln (x))}{x}$

Step 19.2

Divide $3$ by $3$ .

$\frac{{(33 x + 31)}^{\frac{1}{3}} ({(33 x + 31)}^{1} + 44 x \ln (x))}{x}$

Step 19.3

Simplify.

$\frac{{(33 x + 31)}^{\frac{1}{3}} (33 x + 31 + 44 x \ln (x))}{x}$

Step 19.4

Simplify $44 \ln (x)$ by moving $44$ inside the logarithm.

$\frac{{(33 x + 31)}^{\frac{1}{3}} (33 x + 31 + x \ln (x^{44}))}{x}$