Calculus Examples

$f (x) = \ln (| \ln (9 x) |)$

Consider the difference quotient formula.

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

Find the components of the definition.

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

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

$f (x + h) = \ln (| \ln (9 (x + h)) |)$

Simplify the result.

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Apply the distributive property.

$f (x + h) = \ln (| \ln (9 x + 9 h) |)$

The final answer is $\ln (| \ln (9 x + 9 h) |)$ .

$\ln (| \ln (9 x + 9 h) |)$

Find the components of the definition.

$f (x + h) = \ln (| \ln (9 x + 9 h) |)$

$f (x) = \ln (| \ln (9 x) |)$

$f (x + h) = \ln (| \ln (9 x + 9 h) |)$

$f (x) = \ln (| \ln (9 x) |)$

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\ln (| \ln (9 x + 9 h) |) - (\ln (| \ln (9 x) |))}{h}$

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

$\frac{\ln (\frac{| \ln (9 x + 9 h) |}{| \ln (9 x) |})}{h}$