Calculus Examples

$f (x) = \ln (3 x - 1)$

Step 1

Find the first derivative.

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Step 1.1

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 $g (x) = 3 x - 1$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x - 1]$

Step 1.1.2

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

$\frac{1}{u} \frac{d}{d x} [3 x - 1]$

Step 1.1.3

Replace all occurrences of $u$ with $3 x - 1$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Multiply $3$ by $1$ .

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

Step 1.2.5

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$\frac{1}{3 x - 1} (3 + 0)$

Step 1.2.6

Combine fractions.

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Step 1.2.6.1

Add $3$ and $0$ .

$\frac{1}{3 x - 1} \cdot 3$

Step 1.2.6.2

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

$f' (x) = \frac{3}{3 x - 1}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $3 x - 1$ .

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

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $3$ .

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

Step 2.3.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Multiply $3$ by $1$ .

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

Step 2.3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 3 {(3 x - 1)}^{- 2} (3 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $3$ and $0$ .

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

Step 2.3.7.2

Multiply $3$ by $- 3$ .

$- 9 {(3 x - 1)}^{- 2}$

Step 2.4

Simplify.

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Step 2.4.1

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

$- 9 \frac{1}{{(3 x - 1)}^{2}}$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $- 9$ and $\frac{1}{{(3 x - 1)}^{2}}$ .

$\frac{- 9}{{(3 x - 1)}^{2}}$

Step 2.4.2.2

Move the negative in front of the fraction.

$f'' (x) = - \frac{9}{{(3 x - 1)}^{2}}$

Step 3

Find the third derivative.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- \frac{9}{{(3 x - 1)}^{2}}$ with respect to $x$ is $- 9 \frac{d}{d x} [\frac{1}{{(3 x - 1)}^{2}}]$ .

$- 9 \frac{d}{d x} [\frac{1}{{(3 x - 1)}^{2}}]$

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

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

$- 9 \frac{d}{d x} [{({(3 x - 1)}^{2})}^{- 1}]$

Step 3.1.2.2

Multiply the exponents in ${({(3 x - 1)}^{2})}^{- 1}$ .

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Step 3.1.2.2.1

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

$- 9 \frac{d}{d x} [{(3 x - 1)}^{2 \cdot - 1}]$

Step 3.1.2.2.2

Multiply $2$ by $- 1$ .

$- 9 \frac{d}{d x} [{(3 x - 1)}^{- 2}]$

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

$- 9 (\frac{d}{d u} [u^{- 2}] \frac{d}{d x} [3 x - 1])$

Step 3.2.2

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

$- 9 (- 2 u^{- 3} \frac{d}{d x} [3 x - 1])$

Step 3.2.3

Replace all occurrences of $u$ with $3 x - 1$ .

$- 9 (- 2 {(3 x - 1)}^{- 3} \frac{d}{d x} [3 x - 1])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $- 2$ by $- 9$ .

$18 ({(3 x - 1)}^{- 3} \frac{d}{d x} [3 x - 1])$

Step 3.3.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$18 {(3 x - 1)}^{- 3} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])$

Step 3.3.3

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

$18 {(3 x - 1)}^{- 3} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 3.3.4

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

$18 {(3 x - 1)}^{- 3} (3 \cdot 1 + \frac{d}{d x} [- 1])$

Step 3.3.5

Multiply $3$ by $1$ .

$18 {(3 x - 1)}^{- 3} (3 + \frac{d}{d x} [- 1])$

Step 3.3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$18 {(3 x - 1)}^{- 3} (3 + 0)$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Add $3$ and $0$ .

$18 {(3 x - 1)}^{- 3} \cdot 3$

Step 3.3.7.2

Multiply $3$ by $18$ .

$54 {(3 x - 1)}^{- 3}$

Step 3.4

Simplify.

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Step 3.4.1

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

$54 \frac{1}{{(3 x - 1)}^{3}}$

Step 3.4.2

Combine $54$ and $\frac{1}{{(3 x - 1)}^{3}}$ .

$f''' (x) = \frac{54}{{(3 x - 1)}^{3}}$

Step 4

Find the fourth derivative.

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Step 4.1

Differentiate using the Constant Multiple Rule.

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Step 4.1.1

Since $54$ is constant with respect to $x$ , the derivative of $\frac{54}{{(3 x - 1)}^{3}}$ with respect to $x$ is $54 \frac{d}{d x} [\frac{1}{{(3 x - 1)}^{3}}]$ .

$54 \frac{d}{d x} [\frac{1}{{(3 x - 1)}^{3}}]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

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

$54 \frac{d}{d x} [{({(3 x - 1)}^{3})}^{- 1}]$

Step 4.1.2.2

Multiply the exponents in ${({(3 x - 1)}^{3})}^{- 1}$ .

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Step 4.1.2.2.1

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

$54 \frac{d}{d x} [{(3 x - 1)}^{3 \cdot - 1}]$

Step 4.1.2.2.2

Multiply $3$ by $- 1$ .

$54 \frac{d}{d x} [{(3 x - 1)}^{- 3}]$

Step 4.2

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

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Step 4.2.1

To apply the Chain Rule, set $u$ as $3 x - 1$ .

$54 (\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [3 x - 1])$

Step 4.2.2

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

$54 (- 3 u^{- 4} \frac{d}{d x} [3 x - 1])$

Step 4.2.3

Replace all occurrences of $u$ with $3 x - 1$ .

$54 (- 3 {(3 x - 1)}^{- 4} \frac{d}{d x} [3 x - 1])$

Step 4.3

Differentiate.

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Step 4.3.1

Multiply $- 3$ by $54$ .

$- 162 ({(3 x - 1)}^{- 4} \frac{d}{d x} [3 x - 1])$

Step 4.3.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

$- 162 {(3 x - 1)}^{- 4} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])$

Step 4.3.3

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

$- 162 {(3 x - 1)}^{- 4} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 4.3.4

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

$- 162 {(3 x - 1)}^{- 4} (3 \cdot 1 + \frac{d}{d x} [- 1])$

Step 4.3.5

Multiply $3$ by $1$ .

$- 162 {(3 x - 1)}^{- 4} (3 + \frac{d}{d x} [- 1])$

Step 4.3.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$- 162 {(3 x - 1)}^{- 4} (3 + 0)$

Step 4.3.7

Simplify the expression.

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Step 4.3.7.1

Add $3$ and $0$ .

$- 162 {(3 x - 1)}^{- 4} \cdot 3$

Step 4.3.7.2

Multiply $3$ by $- 162$ .

$- 486 {(3 x - 1)}^{- 4}$

Step 4.4

Simplify.

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Step 4.4.1

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

$- 486 \frac{1}{{(3 x - 1)}^{4}}$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Combine $- 486$ and $\frac{1}{{(3 x - 1)}^{4}}$ .

$\frac{- 486}{{(3 x - 1)}^{4}}$

Step 4.4.2.2

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{486}{{(3 x - 1)}^{4}}$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $- \frac{486}{{(3 x - 1)}^{4}}$ .

$- \frac{486}{{(3 x - 1)}^{4}}$