Precalculus Examples

$f (x) = \ln (3 x) + 2.5$

Step 1

Consider the difference quotient formula.

$\frac{f (x + h) - f (x)}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = x + h$ .

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Step 2.1.1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = \ln (3 (x + h)) + 2.5$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = \ln (3 x + 3 h) + 2.5$

Step 2.1.2.2

The final answer is $\ln (3 x + 3 h) + 2.5$ .

$\ln (3 x + 3 h) + 2.5$

Step 2.2

Find the components of the definition.

$f (x + h) = \ln (3 x + 3 h) + 2.5$

$f (x) = \ln (3 x) + 2.5$

$f (x + h) = \ln (3 x + 3 h) + 2.5$

$f (x) = \ln (3 x) + 2.5$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\ln (3 x + 3 h) + 2.5 - (\ln (3 x) + 2.5)}{h}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{\ln (3 x + 3 h) + 2.5 - \ln (3 x) - 1 \cdot 2.5}{h}$

Step 4.2

Multiply $- 1$ by $2.5$ .

$\frac{\ln (3 x + 3 h) + 2.5 - \ln (3 x) - 2.5}{h}$

Step 4.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{\ln (\frac{3 x + 3 h}{3 x}) + 2.5 - 2.5}{h}$

Step 4.4

Subtract $2.5$ from $2.5$ .

$\frac{\ln (\frac{3 x + 3 h}{3 x}) + 0}{h}$

Step 4.5

Add $\ln (\frac{3 x + 3 h}{3 x})$ and $0$ .

$\frac{\ln (\frac{3 x + 3 h}{3 x})}{h}$

Step 4.6

Cancel the common factor of $3 x + 3 h$ and $3$ .

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Step 4.6.1

Factor $3$ out of $3 x$ .

$\frac{\ln (\frac{3 (x) + 3 h}{3 x})}{h}$

Step 4.6.2

Factor $3$ out of $3 h$ .

$\frac{\ln (\frac{3 (x) + 3 h}{3 x})}{h}$

Step 4.6.3

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

$\frac{\ln (\frac{3 (x + h)}{3 x})}{h}$

Step 4.6.4

Cancel the common factors.

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Step 4.6.4.1

Factor $3$ out of $3 x$ .

$\frac{\ln (\frac{3 (x + h)}{3 (x)})}{h}$

Step 4.6.4.2

Cancel the common factor.

$\frac{\ln (\frac{3 (x + h)}{3 x})}{h}$

Step 4.6.4.3

Rewrite the expression.

$\frac{\ln (\frac{x + h}{x})}{h}$

Step 5