Precalculus Examples

Let $f (x) = - \log (4 x) + 5$

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.

$f (x + h) = - \log (4 (x + h)) + 5$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = - \log (4 x + 4 h) + 5$

Step 2.1.2.2

The final answer is $- \log (4 x + 4 h) + 5$ .

$- \log (4 x + 4 h) + 5$

Step 2.2

Find the components of the definition.

$f (x + h) = - \log (4 x + 4 h) + 5$

$f (x) = - \log (4 x) + 5$

$f (x + h) = - \log (4 x + 4 h) + 5$

$f (x) = - \log (4 x) + 5$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \log (4 x + 4 h) + 5 - (- \log (4 x) + 5)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- \log (4 x + 4 h) + 5 - - \log (4 x) - 1 \cdot 5}{h}$

Step 4.1.2

Multiply $- - \log (4 x)$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \log (4 x + 4 h) + 5 + 1 \log (4 x) - 1 \cdot 5}{h}$

Step 4.1.2.2

Multiply $\log (4 x)$ by $1$ .

$\frac{- \log (4 x + 4 h) + 5 + \log (4 x) - 1 \cdot 5}{h}$

Step 4.1.3

Multiply $- 1$ by $5$ .

$\frac{- \log (4 x + 4 h) + 5 + \log (4 x) - 5}{h}$

Step 4.1.4

Subtract $5$ from $5$ .

$\frac{- \log (4 x + 4 h) + 0 + \log (4 x)}{h}$

Step 4.1.5

Add $- \log (4 x + 4 h)$ and $0$ .

$\frac{- \log (4 x + 4 h) + \log (4 x)}{h}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Factor $- 1$ out of $\log (4 x)$ .

$\frac{- \log (4 x + 4 h) - 1 (- \log (4 x))}{h}$

Step 4.2.2

Factor $- 1$ out of $- \log (4 x + 4 h) - 1 (- \log (4 x))$ .

$\frac{- (\log (4 x + 4 h) - \log (4 x))}{h}$

Step 4.2.3

Simplify the expression.

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Step 4.2.3.1

Rewrite $- (\log (4 x + 4 h) - \log (4 x))$ as $- 1 (\log (4 x + 4 h) - \log (4 x))$ .

$\frac{- 1 (\log (4 x + 4 h) - \log (4 x))}{h}$

Step 4.2.3.2

Move the negative in front of the fraction.

$- \frac{\log (4 x + 4 h) - \log (4 x)}{h}$

Step 5