Calculus Examples

$f (x) = \ln (x)$

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.

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

Step 2.1.2

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

$\ln (x + h)$

Step 2.2

Find the components of the definition.

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

$f (x) = \ln (x)$

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

$f (x) = \ln (x)$

Step 3

Plug in the components.

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

Step 4

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

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

Step 5