Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

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 1.1.1.2

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 1.1.2

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

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Step 1.1.2.1

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 1.1.2.2

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 1.1.2.2.1

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 1.1.2.2.2

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 1.1.2.2.3

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

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

Step 1.1.2.3

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 1.1.2.4

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 1.1.2.5

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 1.1.2.6

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 1.1.2.7

Multiply $3$ by $1$ .

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

Step 1.1.2.8

Add $3$ and $0$ .

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

Step 1.1.2.9

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

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

Step 1.1.2.10

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

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Step 1.1.2.10.1

Factor $3$ out of $3$ .

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

Step 1.1.2.10.2

Cancel the common factors.

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Step 1.1.2.10.2.1

Factor $3$ out of $3 x$ .

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

Step 1.1.2.10.2.2

Factor $3$ out of $- 9$ .

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

Step 1.1.2.10.2.3

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

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

Step 1.1.2.10.2.4

Cancel the common factor.

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

Step 1.1.2.10.2.5

Rewrite the expression.

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

Step 1.1.2.11

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

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

Step 1.1.2.12

Move the negative in front of the fraction.

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

Step 1.1.3

Combine terms.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Combine the numerators over the common denominator.

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

Step 1.1.3.3

Subtract $4$ from $- 3$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$x - 7 = 0$

Step 2.3

Add $7$ to both sides of the equation.

$x = 7$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

$x - 3 = 0$

Step 3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4

Evaluate $x - 4 \ln (3 x - 9)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 7$ .

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Step 4.1.1

Substitute $7$ for $x$ .

$(7) - 4 \ln (3 (7) - 9)$

Step 4.1.2

Simplify each term.

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Step 4.1.2.1

Multiply $3$ by $7$ .

$7 - 4 \ln (21 - 9)$

Step 4.1.2.2

Subtract $9$ from $21$ .

$7 - 4 \ln (12)$

Step 4.1.2.3

Simplify $- 4 \ln (12)$ by moving $4$ inside the logarithm.

$7 - \ln (12^{4})$

Step 4.1.2.4

Raise $12$ to the power of $4$ .

$7 - \ln (20736)$

Step 4.2

Evaluate at $x = 3$ .

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Step 4.2.1

Substitute $3$ for $x$ .

$(3) - 4 \ln (3 (3) - 9)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $3$ by $3$ .

$3 - 4 \ln (9 - 9)$

Step 4.2.2.1.2

Subtract $9$ from $9$ .

$3 - 4 \ln (0)$

Step 4.2.2.1.3

The natural logarithm of zero is undefined.

Undefined

$3 - 4 \ln (0)$

Step 4.2.2.2

The natural logarithm of zero is undefined.

Undefined

Step 4.3

List all of the points.

$(7, 7 - \ln (20736))$

Step 5