Calculus Examples

$f (x) = \ln (\sqrt{(6 x - 2) (7 x + 4)})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(6 x - 2) (7 x + 4)}$ as ${((6 x - 2) (7 x + 4))}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\ln ({((6 x - 2) (7 x + 4))}^{\frac{1}{2}})]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = {((6 x - 2) (7 x + 4))}^{\frac{1}{2}}$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as ${((6 x - 2) (7 x + 4))}^{\frac{1}{2}}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}]$

Step 2.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}]$

Step 2.3

Replace all occurrences of $u_{1}$ with ${((6 x - 2) (7 x + 4))}^{\frac{1}{2}}$ .

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} \frac{d}{d x} [{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}]$

Step 3

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 $g (x) = (6 x - 2) (7 x + 4)$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $(6 x - 2) (7 x + 4)$ .

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 3.2

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

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 3.3

Replace all occurrences of $u_{2}$ with $(6 x - 2) (7 x + 4)$ .

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {((6 x - 2) (7 x + 4))}^{\frac{1}{2} - 1} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 4

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

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {((6 x - 2) (7 x + 4))}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 5

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

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {((6 x - 2) (7 x + 4))}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 6

Combine the numerators over the common denominator.

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {((6 x - 2) (7 x + 4))}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {((6 x - 2) (7 x + 4))}^{\frac{1 - 2}{2}} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 7.2

Subtract $2$ from $1$ .

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {((6 x - 2) (7 x + 4))}^{\frac{- 1}{2}} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 8

Move the negative in front of the fraction.

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2} {((6 x - 2) (7 x + 4))}^{- \frac{1}{2}} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 9

Combine $\frac{1}{2}$ and ${((6 x - 2) (7 x + 4))}^{- \frac{1}{2}}$ .

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{{((6 x - 2) (7 x + 4))}^{- \frac{1}{2}}}{2} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 10

Move ${((6 x - 2) (7 x + 4))}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} (\frac{1}{2 {((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} \frac{d}{d x} [(6 x - 2) (7 x + 4)])$

Step 11

Multiply $\frac{1}{2 {((6 x - 2) (7 x + 4))}^{\frac{1}{2}}}$ by $\frac{1}{{((6 x - 2) (7 x + 4))}^{\frac{1}{2}}}$ .

$\frac{1}{2 {((6 x - 2) (7 x + 4))}^{\frac{1}{2}} {((6 x - 2) (7 x + 4))}^{\frac{1}{2}}} \frac{d}{d x} [(6 x - 2) (7 x + 4)]$

Step 12

Multiply ${((6 x - 2) (7 x + 4))}^{\frac{1}{2}}$ by ${((6 x - 2) (7 x + 4))}^{\frac{1}{2}}$ by adding the exponents.

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Step 12.1

Move ${((6 x - 2) (7 x + 4))}^{\frac{1}{2}}$ .

$\frac{1}{2 ({((6 x - 2) (7 x + 4))}^{\frac{1}{2}} {((6 x - 2) (7 x + 4))}^{\frac{1}{2}})} \frac{d}{d x} [(6 x - 2) (7 x + 4)]$

Step 12.2

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

$\frac{1}{2 {((6 x - 2) (7 x + 4))}^{\frac{1}{2} + \frac{1}{2}}} \frac{d}{d x} [(6 x - 2) (7 x + 4)]$

Step 12.3

Combine the numerators over the common denominator.

$\frac{1}{2 {((6 x - 2) (7 x + 4))}^{\frac{1 + 1}{2}}} \frac{d}{d x} [(6 x - 2) (7 x + 4)]$

Step 12.4

Add $1$ and $1$ .

$\frac{1}{2 {((6 x - 2) (7 x + 4))}^{\frac{2}{2}}} \frac{d}{d x} [(6 x - 2) (7 x + 4)]$

Step 12.5

Divide $2$ by $2$ .

$\frac{1}{2 {((6 x - 2) (7 x + 4))}^{1}} \frac{d}{d x} [(6 x - 2) (7 x + 4)]$

Step 13

Simplify $2 {((6 x - 2) (7 x + 4))}^{1}$ .

$\frac{1}{2 ((6 x - 2) (7 x + 4))} \frac{d}{d x} [(6 x - 2) (7 x + 4)]$

Step 14

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 $f (x) = 6 x - 2$ and $g (x) = 7 x + 4$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} ((6 x - 2) \frac{d}{d x} [7 x + 4] + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15

Differentiate.

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Step 15.1

By the Sum Rule, the derivative of $7 x + 4$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [4]$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} ((6 x - 2) (\frac{d}{d x} [7 x] + \frac{d}{d x} [4]) + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} ((6 x - 2) (7 \frac{d}{d x} [x] + \frac{d}{d x} [4]) + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15.3

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

$\frac{1}{2 (6 x - 2) (7 x + 4)} ((6 x - 2) (7 \cdot 1 + \frac{d}{d x} [4]) + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15.4

Multiply $7$ by $1$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} ((6 x - 2) (7 + \frac{d}{d x} [4]) + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} ((6 x - 2) (7 + 0) + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15.6

Simplify the expression.

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Step 15.6.1

Add $7$ and $0$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} ((6 x - 2) \cdot 7 + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15.6.2

Move $7$ to the left of $6 x - 2$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 \cdot (6 x - 2) + (7 x + 4) \frac{d}{d x} [6 x - 2])$

Step 15.7

By the Sum Rule, the derivative of $6 x - 2$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 2]$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 (6 x - 2) + (7 x + 4) (\frac{d}{d x} [6 x] + \frac{d}{d x} [- 2]))$

Step 15.8

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 (6 x - 2) + (7 x + 4) (6 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]))$

Step 15.9

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

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 (6 x - 2) + (7 x + 4) (6 \cdot 1 + \frac{d}{d x} [- 2]))$

Step 15.10

Multiply $6$ by $1$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 (6 x - 2) + (7 x + 4) (6 + \frac{d}{d x} [- 2]))$

Step 15.11

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 (6 x - 2) + (7 x + 4) (6 + 0))$

Step 15.12

Simplify the expression.

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Step 15.12.1

Add $6$ and $0$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 (6 x - 2) + (7 x + 4) \cdot 6)$

Step 15.12.2

Move $6$ to the left of $7 x + 4$ .

$\frac{1}{2 (6 x - 2) (7 x + 4)} (7 (6 x - 2) + 6 \cdot (7 x + 4))$

Step 16

Simplify.

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Step 16.1

Apply the distributive property.

$\frac{1}{(2 (6 x) + 2 \cdot - 2) (7 x + 4)} (7 (6 x - 2) + 6 (7 x + 4))$

Step 16.2

Apply the distributive property.

$\frac{1}{(2 (6 x) + 2 \cdot - 2) (7 x + 4)} (7 (6 x) + 7 \cdot - 2 + 6 (7 x + 4))$

Step 16.3

Apply the distributive property.

$\frac{1}{(2 (6 x) + 2 \cdot - 2) (7 x + 4)} (7 (6 x) + 7 \cdot - 2 + 6 (7 x) + 6 \cdot 4)$

Step 16.4

Combine terms.

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Step 16.4.1

Multiply $6$ by $2$ .

$\frac{1}{(12 x + 2 \cdot - 2) (7 x + 4)} (7 (6 x) + 7 \cdot - 2 + 6 (7 x) + 6 \cdot 4)$

Step 16.4.2

Multiply $2$ by $- 2$ .

$\frac{1}{(12 x - 4) (7 x + 4)} (7 (6 x) + 7 \cdot - 2 + 6 (7 x) + 6 \cdot 4)$

Step 16.4.3

Multiply $6$ by $7$ .

$\frac{1}{(12 x - 4) (7 x + 4)} (42 x + 7 \cdot - 2 + 6 (7 x) + 6 \cdot 4)$

Step 16.4.4

Multiply $7$ by $- 2$ .

$\frac{1}{(12 x - 4) (7 x + 4)} (42 x - 14 + 6 (7 x) + 6 \cdot 4)$

Step 16.4.5

Multiply $7$ by $6$ .

$\frac{1}{(12 x - 4) (7 x + 4)} (42 x - 14 + 42 x + 6 \cdot 4)$

Step 16.4.6

Multiply $6$ by $4$ .

$\frac{1}{(12 x - 4) (7 x + 4)} (42 x - 14 + 42 x + 24)$

Step 16.4.7

Add $42 x$ and $42 x$ .

$\frac{1}{(12 x - 4) (7 x + 4)} (84 x - 14 + 24)$

Step 16.4.8

Add $- 14$ and $24$ .

$\frac{1}{(12 x - 4) (7 x + 4)} (84 x + 10)$

Step 16.5

Reorder the factors of $\frac{1}{(12 x - 4) (7 x + 4)} (84 x + 10)$ .

$(84 x + 10) \frac{1}{(12 x - 4) (7 x + 4)}$

Step 16.6

Factor $4$ out of $12 x - 4$ .

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Step 16.6.1

Factor $4$ out of $12 x$ .

$(84 x + 10) \frac{1}{(4 (3 x) - 4) (7 x + 4)}$

Step 16.6.2

Factor $4$ out of $- 4$ .

$(84 x + 10) \frac{1}{(4 (3 x) + 4 (- 1)) (7 x + 4)}$

Step 16.6.3

Factor $4$ out of $4 (3 x) + 4 (- 1)$ .

$(84 x + 10) \frac{1}{4 (3 x - 1) (7 x + 4)}$

Step 16.7

Multiply $84 x + 10$ by $\frac{1}{4 (3 x - 1) (7 x + 4)}$ .

$\frac{84 x + 10}{4 (3 x - 1) (7 x + 4)}$

Step 16.8

Cancel the common factor of $84 x + 10$ and $4$ .

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Step 16.8.1

Factor $2$ out of $84 x$ .

$\frac{2 (42 x) + 10}{4 (3 x - 1) (7 x + 4)}$

Step 16.8.2

Factor $2$ out of $10$ .

$\frac{2 (42 x) + 2 (5)}{4 (3 x - 1) (7 x + 4)}$

Step 16.8.3

Factor $2$ out of $2 (42 x) + 2 (5)$ .

$\frac{2 (42 x + 5)}{4 (3 x - 1) (7 x + 4)}$

Step 16.8.4

Cancel the common factors.

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Step 16.8.4.1

Factor $2$ out of $4 (3 x - 1) (7 x + 4)$ .

$\frac{2 (42 x + 5)}{2 (2 (3 x - 1) (7 x + 4))}$

Step 16.8.4.2

Cancel the common factor.

$\frac{2 (42 x + 5)}{2 (2 (3 x - 1) (7 x + 4))}$

Step 16.8.4.3

Rewrite the expression.

$\frac{42 x + 5}{2 (3 x - 1) (7 x + 4)}$