Calculus Examples

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

Step 1

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

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

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 $x - 4 \ln (3 x - 4)$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4 \ln (3 x - 4)]$ .

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

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 $n = 1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

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

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 $g (x) = 3 x - 4$ .

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Step 2.1.2.2.1

To apply the Chain Rule, set $u$ as $3 x - 4$ .

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

Step 2.1.2.2.2

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

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

Step 2.1.2.2.3

Replace all occurrences of $u$ with $3 x - 4$ .

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

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 $n = 1$ .

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

Step 2.1.2.6

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

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

Step 2.1.2.7

Multiply $3$ by $1$ .

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

Step 2.1.2.8

Add $3$ and $0$ .

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

Step 2.1.2.9

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

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

Step 2.1.2.10

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

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

Step 2.1.2.11

Multiply $- 4$ by $3$ .

$1 + \frac{- 12}{3 x - 4}$

Step 2.1.2.12

Move the negative in front of the fraction.

$1 - \frac{12}{3 x - 4}$

Step 2.1.3

Combine terms.

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Step 2.1.3.1

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

$\frac{3 x - 4}{3 x - 4} - \frac{12}{3 x - 4}$

Step 2.1.3.2

Combine the numerators over the common denominator.

$\frac{3 x - 4 - 12}{3 x - 4}$

Step 2.1.3.3

Subtract $12$ from $- 4$ .

$f' (x) = \frac{3 x - 16}{3 x - 4}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{3 x - 16}{3 x - 4}$ .

$\frac{3 x - 16}{3 x - 4}$

Step 3

Set the first derivative equal to $0$ then solve the equation $\frac{3 x - 16}{3 x - 4} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$\frac{3 x - 16}{3 x - 4} = 0$

Step 3.2

Set the numerator equal to zero.

$3 x - 16 = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

Add $16$ to both sides of the equation.

$3 x = 16$

Step 3.3.2

Divide each term in $3 x = 16$ by $3$ and simplify.

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Step 3.3.2.1

Divide each term in $3 x = 16$ by $3$ .

$\frac{3 x}{3} = \frac{16}{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{16}{3}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{3}$

Step 4

The values which make the derivative equal to $0$ are $\frac{16}{3}$ .

$\frac{16}{3}$

Step 5

Find where the derivative is undefined.

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Step 5.1

Set the denominator in $\frac{3 x - 16}{3 x - 4}$ equal to $0$ to find where the expression is undefined.

$3 x - 4 = 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 5.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 x = 4$ by $3$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6

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

$(- \infty, \frac{4}{3}) \cup (\frac{4}{3}, \frac{16}{3}) \cup (\frac{16}{3}, \infty)$

Step 7

Exclude the intervals that are not in the domain.

$(\frac{4}{3}, \frac{16}{3})$

Step 8

Substitute a value from the interval $(1.3333334, \frac{16}{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $3.33333345$ in the expression.

$f' (3.33333345) = \frac{3 (3.33333345) - 16}{3 (3.33333345) - 4}$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify the numerator.

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Step 8.2.1.1

Multiply $3$ by $3.33333345$ .

$f' (3.33333345) = \frac{10.00000035 - 16}{3 (3.33333345) - 4}$

Step 8.2.1.2

Subtract $16$ from $10.00000035$ .

$f' (3.33333345) = \frac{- 5.99999965}{3 (3.33333345) - 4}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Multiply $3$ by $3.33333345$ .

$f' (3.33333345) = \frac{- 5.99999965}{10.00000035 - 4}$

Step 8.2.2.2

Subtract $4$ from $10.00000035$ .

$f' (3.33333345) = \frac{- 5.99999965}{6.00000035}$

Step 8.2.3

Divide $- 5.99999965$ by $6.00000035$ .

$f' (3.33333345) = - 0.99999988$

Step 8.2.4

The final answer is $- 0.99999988$ .

$- 0.99999988$

Step 8.3

At $x = 3.33333345$ the derivative is $- 0.99999988$ . Since this is negative, the function is decreasing on $(1.3333334, \frac{16}{3})$ .

Decreasing on $(\frac{4}{3}, \frac{16}{3})$ since $f' (x) < 0$

Step 9

Exclude the intervals that are not in the domain.

$(\frac{4}{3}, \frac{16}{3}), (\frac{16}{3}, \infty)$

Step 10

Substitute a value from the interval $(\frac{16}{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $6.3333335$ in the expression.

$f' (6.3333335) = \frac{3 (6.3333335) - 16}{3 (6.3333335) - 4}$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify the numerator.

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Step 10.2.1.1

Multiply $3$ by $6.3333335$ .

$f' (6.3333335) = \frac{19.0000005 - 16}{3 (6.3333335) - 4}$

Step 10.2.1.2

Subtract $16$ from $19.0000005$ .

$f' (6.3333335) = \frac{3.0000005}{3 (6.3333335) - 4}$

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Multiply $3$ by $6.3333335$ .

$f' (6.3333335) = \frac{3.0000005}{19.0000005 - 4}$

Step 10.2.2.2

Subtract $4$ from $19.0000005$ .

$f' (6.3333335) = \frac{3.0000005}{15.0000005}$

Step 10.2.3

Divide $3.0000005$ by $15.0000005$ .

$f' (6.3333335) = 0.20000002$

Step 10.2.4

The final answer is $0.20000002$ .

$0.20000002$

Step 10.3

At $x = 6.3333335$ the derivative is $0.20000002$ . Since this is positive, the function is increasing on $(\frac{16}{3}, \infty)$ .

Increasing on $(\frac{16}{3}, \infty)$ since $f' (x) > 0$

Step 11

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

Increasing on: $(\frac{16}{3}, \infty)$

Decreasing on: $(\frac{4}{3}, \frac{16}{3})$

Step 12