Calculus Examples

$y = x - 4 \ln (3 x - 9)$

Step 1

Write $y = x - 4 \ln (3 x - 9)$ as a function.

$f (x) = x - 4 \ln (3 x - 9)$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

Differentiate.

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Step 2.1.1.1

By the Sum Rule, the derivative of $x - 4 \ln (3 x - 9)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4 \ln (3 x - 9)]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 4 \ln (3 x - 9)]$

Step 2.1.1.2

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

$1 + \frac{d}{d x} [- 4 \ln (3 x - 9)]$

Step 2.1.2

Evaluate $\frac{d}{d x} [- 4 \ln (3 x - 9)]$ .

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Step 2.1.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \ln (3 x - 9)$ with respect to $x$ is $- 4 \frac{d}{d x} [\ln (3 x - 9)]$ .

$1 - 4 \frac{d}{d x} [\ln (3 x - 9)]$

Step 2.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x - 9$ .

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Step 2.1.2.2.1

To apply the Chain Rule, set $u$ as $3 x - 9$ .

$1 - 4 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x - 9])$

Step 2.1.2.2.2

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

$1 - 4 (\frac{1}{u} \frac{d}{d x} [3 x - 9])$

Step 2.1.2.2.3

Replace all occurrences of $u$ with $3 x - 9$ .

$1 - 4 (\frac{1}{3 x - 9} \frac{d}{d x} [3 x - 9])$

Step 2.1.2.3

By the Sum Rule, the derivative of $3 x - 9$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$ .

$1 - 4 (\frac{1}{3 x - 9} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]))$

Step 2.1.2.4

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

$1 - 4 (\frac{1}{3 x - 9} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]))$

Step 2.1.2.5

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

$1 - 4 (\frac{1}{3 x - 9} (3 \cdot 1 + \frac{d}{d x} [- 9]))$

Step 2.1.2.6

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$1 - 4 (\frac{1}{3 x - 9} (3 \cdot 1 + 0))$

Step 2.1.2.7

Multiply $3$ by $1$ .

$1 - 4 (\frac{1}{3 x - 9} (3 + 0))$

Step 2.1.2.8

Add $3$ and $0$ .

$1 - 4 (\frac{1}{3 x - 9} \cdot 3)$

Step 2.1.2.9

Combine $\frac{1}{3 x - 9}$ and $3$ .

$1 - 4 \frac{3}{3 x - 9}$

Step 2.1.2.10

Cancel the common factor of $3$ and $3 x - 9$ .

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Step 2.1.2.10.1

Factor $3$ out of $3$ .

$1 - 4 \frac{3 \cdot 1}{3 x - 9}$

Step 2.1.2.10.2

Cancel the common factors.

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Step 2.1.2.10.2.1

Factor $3$ out of $3 x$ .

$1 - 4 \frac{3 \cdot 1}{3 (x) - 9}$

Step 2.1.2.10.2.2

Factor $3$ out of $- 9$ .

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

Step 2.1.2.10.2.3

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

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

Step 2.1.2.10.2.4

Cancel the common factor.

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

Step 2.1.2.10.2.5

Rewrite the expression.

$1 - 4 \frac{1}{x - 3}$

Step 2.1.2.11

Combine $- 4$ and $\frac{1}{x - 3}$ .

$1 + \frac{- 4}{x - 3}$

Step 2.1.2.12

Move the negative in front of the fraction.

$1 - \frac{4}{x - 3}$

Step 2.1.3

Combine terms.

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Step 2.1.3.1

Write $1$ as a fraction with a common denominator.

$\frac{x - 3}{x - 3} - \frac{4}{x - 3}$

Step 2.1.3.2

Combine the numerators over the common denominator.

$\frac{x - 3 - 4}{x - 3}$

Step 2.1.3.3

Subtract $4$ from $- 3$ .

$f' (x) = \frac{x - 7}{x - 3}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{x - 7}{x - 3}$ .

$\frac{x - 7}{x - 3}$

Step 3

Set the first derivative equal to $0$ then solve the equation $\frac{x - 7}{x - 3} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$\frac{x - 7}{x - 3} = 0$

Step 3.2

Set the numerator equal to zero.

$x - 7 = 0$

Step 3.3

Add $7$ to both sides of the equation.

$x = 7$

Step 4

The values which make the derivative equal to $0$ are $7$ .

$7$

Step 5

Find where the derivative is undefined.

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Step 5.1

Set the denominator in $\frac{x - 7}{x - 3}$ equal to $0$ to find where the expression is undefined.

$x - 3 = 0$

Step 5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 3) \cup (3, 7) \cup (7, \infty)$

Step 7

Exclude the intervals that are not in the domain.

$(3, 7)$

Step 8

Substitute a value from the interval $(3, 7)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $5$ in the expression.

$f' (5) = \frac{(5) - 7}{(5) - 3}$

Step 8.2

Simplify the result.

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Step 8.2.1

Subtract $7$ from $5$ .

$f' (5) = \frac{- 2}{5 - 3}$

Step 8.2.2

Subtract $3$ from $5$ .

$f' (5) = \frac{- 2}{2}$

Step 8.2.3

Divide $- 2$ by $2$ .

$f' (5) = - 1$

Step 8.2.4

The final answer is $- 1$ .

$- 1$

Step 8.3

At $x = 5$ the derivative is $- 1$ . Since this is negative, the function is decreasing on $(3, 7)$ .

Decreasing on $(3, 7)$ since $f' (x) < 0$

Step 9

Exclude the intervals that are not in the domain.

$(3, 7), (7, \infty)$

Step 10

Substitute a value from the interval $(7, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = \frac{(8) - 7}{(8) - 3}$

Step 10.2

Simplify the result.

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Step 10.2.1

Subtract $7$ from $8$ .

$f' (8) = \frac{1}{8 - 3}$

Step 10.2.2

Subtract $3$ from $8$ .

$f' (8) = \frac{1}{5}$

Step 10.2.3

The final answer is $\frac{1}{5}$ .

$\frac{1}{5}$

Step 10.3

At $x = 8$ the derivative is $\frac{1}{5}$ . Since this is positive, the function is increasing on $(7, \infty)$ .

Increasing on $(7, \infty)$ since $f' (x) > 0$

Step 11

List the intervals on which the function is increasing and decreasing.

Increasing on: $(7, \infty)$

Decreasing on: $(3, 7)$

Step 12