Calculus Examples

$y = \sqrt{4 x - 1}$

Step 1

Find the first derivative.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 1}$ as ${(4 x - 1)}^{\frac{1}{2}}$ .

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

Step 1.2

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 $g (x) = 4 x - 1$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $4 x - 1$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $4 x - 1$ .

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

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

Step 1.6.2

Subtract $2$ from $1$ .

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

Step 1.7

Combine fractions.

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Step 1.7.1

Move the negative in front of the fraction.

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

Step 1.7.2

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

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

Step 1.7.3

Move ${(4 x - 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.8

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

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

Step 1.9

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

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

Step 1.10

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

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

Step 1.11

Multiply $4$ by $1$ .

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

Step 1.12

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

$\frac{1}{2 {(4 x - 1)}^{\frac{1}{2}}} (4 + 0)$

Step 1.13

Simplify terms.

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Step 1.13.1

Add $4$ and $0$ .

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

Step 1.13.2

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

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

Step 1.13.3

Factor $2$ out of $4$ .

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

Step 1.14

Cancel the common factors.

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Step 1.14.1

Factor $2$ out of $2 {(4 x - 1)}^{\frac{1}{2}}$ .

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

Step 1.14.2

Cancel the common factor.

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

Step 1.14.3

Rewrite the expression.

$f' (x) = \frac{2}{{(4 x - 1)}^{\frac{1}{2}}}$

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 $2$ is constant with respect to $x$ , the derivative of $\frac{2}{{(4 x - 1)}^{\frac{1}{2}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{{(4 x - 1)}^{\frac{1}{2}}}]$ .

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{{(4 x - 1)}^{\frac{1}{2}}}$ as ${({(4 x - 1)}^{\frac{1}{2}})}^{- 1}$ .

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

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

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

Step 2.1.2.2.3

Move the negative in front of the fraction.

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

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 $f (x) = x^{- \frac{1}{2}}$ and $g (x) = 4 x - 1$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $4 x - 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $4 x - 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

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

Step 2.6.2

Subtract $2$ from $- 1$ .

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

Step 2.7

Move the negative in front of the fraction.

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

Step 2.8

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

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

Step 2.9

Simplify the expression.

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Step 2.9.1

Move ${(4 x - 1)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.9.2

Multiply $- 1$ by $2$ .

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

Step 2.10

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

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

Step 2.11

Factor $2$ out of $- 2$ .

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

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $2$ out of $2 {(4 x - 1)}^{\frac{3}{2}}$ .

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

Step 2.12.2

Cancel the common factor.

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

Step 2.12.3

Rewrite the expression.

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

Step 2.13

Move the negative in front of the fraction.

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

Step 2.14

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

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

Step 2.15

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

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

Step 2.16

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

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

Step 2.17

Multiply $4$ by $1$ .

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

Step 2.18

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

$- \frac{1}{{(4 x - 1)}^{\frac{3}{2}}} (4 + 0)$

Step 2.19

Combine fractions.

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Step 2.19.1

Add $4$ and $0$ .

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

Step 2.19.2

Multiply $4$ by $- 1$ .

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

Step 2.19.3

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

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

Step 2.19.4

Move the negative in front of the fraction.

$f'' (x) = - \frac{4}{{(4 x - 1)}^{\frac{3}{2}}}$

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 $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4}{{(4 x - 1)}^{\frac{3}{2}}}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{1}{{(4 x - 1)}^{\frac{3}{2}}}]$ .

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

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(4 x - 1)}^{\frac{3}{2}}}$ as ${({(4 x - 1)}^{\frac{3}{2}})}^{- 1}$ .

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

Step 3.1.2.2

Multiply the exponents in ${({(4 x - 1)}^{\frac{3}{2}})}^{- 1}$ .

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.1.2.2.2.1

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

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

Step 3.1.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.1.2.2.3

Move the negative in front of the fraction.

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

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 $f (x) = x^{- \frac{3}{2}}$ and $g (x) = 4 x - 1$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $4 x - 1$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $4 x - 1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $- 3$ .

$- 4 (- \frac{3}{2} {(4 x - 1)}^{\frac{- 5}{2}} \frac{d}{d x} [4 x - 1])$

Step 3.7

Move the negative in front of the fraction.

$- 4 (- \frac{3}{2} {(4 x - 1)}^{- \frac{5}{2}} \frac{d}{d x} [4 x - 1])$

Step 3.8

Combine ${(4 x - 1)}^{- \frac{5}{2}}$ and $\frac{3}{2}$ .

$- 4 (- \frac{{(4 x - 1)}^{- \frac{5}{2}} \cdot 3}{2} \frac{d}{d x} [4 x - 1])$

Step 3.9

Simplify the expression.

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Step 3.9.1

Move $3$ to the left of ${(4 x - 1)}^{- \frac{5}{2}}$ .

$- 4 (- \frac{3 \cdot {(4 x - 1)}^{- \frac{5}{2}}}{2} \frac{d}{d x} [4 x - 1])$

Step 3.9.2

Move ${(4 x - 1)}^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 4 (- \frac{3}{2 {(4 x - 1)}^{\frac{5}{2}}} \frac{d}{d x} [4 x - 1])$

Step 3.9.3

Multiply $- 1$ by $- 4$ .

$4 (\frac{3}{2 {(4 x - 1)}^{\frac{5}{2}}} \frac{d}{d x} [4 x - 1])$

Step 3.10

Combine $\frac{3}{2 {(4 x - 1)}^{\frac{5}{2}}}$ and $4$ .

$\frac{3 \cdot 4}{2 {(4 x - 1)}^{\frac{5}{2}}} \frac{d}{d x} [4 x - 1]$

Step 3.11

Multiply $3$ by $4$ .

$\frac{12}{2 {(4 x - 1)}^{\frac{5}{2}}} \frac{d}{d x} [4 x - 1]$

Step 3.12

Factor $2$ out of $12$ .

$\frac{2 \cdot 6}{2 {(4 x - 1)}^{\frac{5}{2}}} \frac{d}{d x} [4 x - 1]$

Step 3.13

Cancel the common factors.

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Step 3.13.1

Factor $2$ out of $2 {(4 x - 1)}^{\frac{5}{2}}$ .

$\frac{2 \cdot 6}{2 ({(4 x - 1)}^{\frac{5}{2}})} \frac{d}{d x} [4 x - 1]$

Step 3.13.2

Cancel the common factor.

$\frac{2 \cdot 6}{2 {(4 x - 1)}^{\frac{5}{2}}} \frac{d}{d x} [4 x - 1]$

Step 3.13.3

Rewrite the expression.

$\frac{6}{{(4 x - 1)}^{\frac{5}{2}}} \frac{d}{d x} [4 x - 1]$

Step 3.14

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

$\frac{6}{{(4 x - 1)}^{\frac{5}{2}}} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1])$

Step 3.15

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

$\frac{6}{{(4 x - 1)}^{\frac{5}{2}}} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 3.16

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

$\frac{6}{{(4 x - 1)}^{\frac{5}{2}}} (4 \cdot 1 + \frac{d}{d x} [- 1])$

Step 3.17

Multiply $4$ by $1$ .

$\frac{6}{{(4 x - 1)}^{\frac{5}{2}}} (4 + \frac{d}{d x} [- 1])$

Step 3.18

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

$\frac{6}{{(4 x - 1)}^{\frac{5}{2}}} (4 + 0)$

Step 3.19

Combine fractions.

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Step 3.19.1

Add $4$ and $0$ .

$\frac{6}{{(4 x - 1)}^{\frac{5}{2}}} \cdot 4$

Step 3.19.2

Combine $\frac{6}{{(4 x - 1)}^{\frac{5}{2}}}$ and $4$ .

$\frac{6 \cdot 4}{{(4 x - 1)}^{\frac{5}{2}}}$

Step 3.19.3

Multiply $6$ by $4$ .

$f''' (x) = \frac{24}{{(4 x - 1)}^{\frac{5}{2}}}$

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 $24$ is constant with respect to $x$ , the derivative of $\frac{24}{{(4 x - 1)}^{\frac{5}{2}}}$ with respect to $x$ is $24 \frac{d}{d x} [\frac{1}{{(4 x - 1)}^{\frac{5}{2}}}]$ .

$24 \frac{d}{d x} [\frac{1}{{(4 x - 1)}^{\frac{5}{2}}}]$

Step 4.1.2

Apply basic rules of exponents.

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Step 4.1.2.1

Rewrite $\frac{1}{{(4 x - 1)}^{\frac{5}{2}}}$ as ${({(4 x - 1)}^{\frac{5}{2}})}^{- 1}$ .

$24 \frac{d}{d x} [{({(4 x - 1)}^{\frac{5}{2}})}^{- 1}]$

Step 4.1.2.2

Multiply the exponents in ${({(4 x - 1)}^{\frac{5}{2}})}^{- 1}$ .

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Step 4.1.2.2.1

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

$24 \frac{d}{d x} [{(4 x - 1)}^{\frac{5}{2} \cdot - 1}]$

Step 4.1.2.2.2

Multiply $\frac{5}{2} \cdot - 1$ .

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Step 4.1.2.2.2.1

Combine $\frac{5}{2}$ and $- 1$ .

$24 \frac{d}{d x} [{(4 x - 1)}^{\frac{5 \cdot - 1}{2}}]$

Step 4.1.2.2.2.2

Multiply $5$ by $- 1$ .

$24 \frac{d}{d x} [{(4 x - 1)}^{\frac{- 5}{2}}]$

Step 4.1.2.2.3

Move the negative in front of the fraction.

$24 \frac{d}{d x} [{(4 x - 1)}^{- \frac{5}{2}}]$

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 $f (x) = x^{- \frac{5}{2}}$ and $g (x) = 4 x - 1$ .

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Step 4.2.1

To apply the Chain Rule, set $u$ as $4 x - 1$ .

$24 (\frac{d}{d u} [u^{- \frac{5}{2}}] \frac{d}{d x} [4 x - 1])$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{5}{2}$ .

$24 (- \frac{5}{2} u^{- \frac{5}{2} - 1} \frac{d}{d x} [4 x - 1])$

Step 4.2.3

Replace all occurrences of $u$ with $4 x - 1$ .

$24 (- \frac{5}{2} {(4 x - 1)}^{- \frac{5}{2} - 1} \frac{d}{d x} [4 x - 1])$

Step 4.3

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

$24 (- \frac{5}{2} {(4 x - 1)}^{- \frac{5}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [4 x - 1])$

Step 4.4

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

$24 (- \frac{5}{2} {(4 x - 1)}^{- \frac{5}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [4 x - 1])$

Step 4.5

Combine the numerators over the common denominator.

$24 (- \frac{5}{2} {(4 x - 1)}^{\frac{- 5 - 1 \cdot 2}{2}} \frac{d}{d x} [4 x - 1])$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 1$ by $2$ .

$24 (- \frac{5}{2} {(4 x - 1)}^{\frac{- 5 - 2}{2}} \frac{d}{d x} [4 x - 1])$

Step 4.6.2

Subtract $2$ from $- 5$ .

$24 (- \frac{5}{2} {(4 x - 1)}^{\frac{- 7}{2}} \frac{d}{d x} [4 x - 1])$

Step 4.7

Move the negative in front of the fraction.

$24 (- \frac{5}{2} {(4 x - 1)}^{- \frac{7}{2}} \frac{d}{d x} [4 x - 1])$

Step 4.8

Combine ${(4 x - 1)}^{- \frac{7}{2}}$ and $\frac{5}{2}$ .

$24 (- \frac{{(4 x - 1)}^{- \frac{7}{2}} \cdot 5}{2} \frac{d}{d x} [4 x - 1])$

Step 4.9

Simplify the expression.

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Step 4.9.1

Move $5$ to the left of ${(4 x - 1)}^{- \frac{7}{2}}$ .

$24 (- \frac{5 \cdot {(4 x - 1)}^{- \frac{7}{2}}}{2} \frac{d}{d x} [4 x - 1])$

Step 4.9.2

Move ${(4 x - 1)}^{- \frac{7}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$24 (- \frac{5}{2 {(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1])$

Step 4.9.3

Multiply $- 1$ by $24$ .

$- 24 (\frac{5}{2 {(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1])$

Step 4.10

Combine $\frac{5}{2 {(4 x - 1)}^{\frac{7}{2}}}$ and $- 24$ .

$\frac{5 \cdot - 24}{2 {(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1]$

Step 4.11

Multiply $5$ by $- 24$ .

$\frac{- 120}{2 {(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1]$

Step 4.12

Factor $2$ out of $- 120$ .

$\frac{2 \cdot - 60}{2 {(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1]$

Step 4.13

Cancel the common factors.

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Step 4.13.1

Factor $2$ out of $2 {(4 x - 1)}^{\frac{7}{2}}$ .

$\frac{2 \cdot - 60}{2 ({(4 x - 1)}^{\frac{7}{2}})} \frac{d}{d x} [4 x - 1]$

Step 4.13.2

Cancel the common factor.

$\frac{2 \cdot - 60}{2 {(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1]$

Step 4.13.3

Rewrite the expression.

$\frac{- 60}{{(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1]$

Step 4.14

Move the negative in front of the fraction.

$- \frac{60}{{(4 x - 1)}^{\frac{7}{2}}} \frac{d}{d x} [4 x - 1]$

Step 4.15

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

$- \frac{60}{{(4 x - 1)}^{\frac{7}{2}}} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1])$

Step 4.16

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

$- \frac{60}{{(4 x - 1)}^{\frac{7}{2}}} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 4.17

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

$- \frac{60}{{(4 x - 1)}^{\frac{7}{2}}} (4 \cdot 1 + \frac{d}{d x} [- 1])$

Step 4.18

Multiply $4$ by $1$ .

$- \frac{60}{{(4 x - 1)}^{\frac{7}{2}}} (4 + \frac{d}{d x} [- 1])$

Step 4.19

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

$- \frac{60}{{(4 x - 1)}^{\frac{7}{2}}} (4 + 0)$

Step 4.20

Combine fractions.

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Step 4.20.1

Add $4$ and $0$ .

$- \frac{60}{{(4 x - 1)}^{\frac{7}{2}}} \cdot 4$

Step 4.20.2

Multiply $4$ by $- 1$ .

$- 4 \frac{60}{{(4 x - 1)}^{\frac{7}{2}}}$

Step 4.20.3

Combine $- 4$ and $\frac{60}{{(4 x - 1)}^{\frac{7}{2}}}$ .

$\frac{- 4 \cdot 60}{{(4 x - 1)}^{\frac{7}{2}}}$

Step 4.20.4

Simplify the expression.

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Step 4.20.4.1

Multiply $- 4$ by $60$ .

$\frac{- 240}{{(4 x - 1)}^{\frac{7}{2}}}$

Step 4.20.4.2

Move the negative in front of the fraction.

$f^{4} (x) = - \frac{240}{{(4 x - 1)}^{\frac{7}{2}}}$