Precalculus Examples

$a (t) = \frac{1}{t + 8}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$a (t + h) = \frac{1}{(t + h) + 8}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$a (t + h) = \frac{1}{t + h + 8}$

Step 2.1.2.2

The final answer is $\frac{1}{t + h + 8}$ .

$\frac{1}{t + h + 8}$

Step 2.2

Find the components of the definition.

$a (t + h) = \frac{1}{t + h + 8}$

$a (t) = \frac{1}{t + 8}$

$a (t + h) = \frac{1}{t + h + 8}$

$a (t) = \frac{1}{t + 8}$

Step 3

Plug in the components.

$\frac{a (t + h) - a t}{h} = \frac{\frac{1}{t + h + 8} - (\frac{1}{t + 8})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1}{t + h + 8}$ as a fraction with a common denominator, multiply by $\frac{t + 8}{t + 8}$ .

$\frac{\frac{1}{t + h + 8} \cdot \frac{t + 8}{t + 8} - \frac{1}{t + 8}}{h}$

Step 4.1.2

To write $- \frac{1}{t + 8}$ as a fraction with a common denominator, multiply by $\frac{t + h + 8}{t + h + 8}$ .

$\frac{\frac{1}{t + h + 8} \cdot \frac{t + 8}{t + 8} - \frac{1}{t + 8} \cdot \frac{t + h + 8}{t + h + 8}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(t + h + 8) (t + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{1}{t + h + 8}$ by $\frac{t + 8}{t + 8}$ .

$\frac{\frac{t + 8}{(t + h + 8) (t + 8)} - \frac{1}{t + 8} \cdot \frac{t + h + 8}{t + h + 8}}{h}$

Step 4.1.3.2

Multiply $\frac{1}{t + 8}$ by $\frac{t + h + 8}{t + h + 8}$ .

$\frac{\frac{t + 8}{(t + h + 8) (t + 8)} - \frac{t + h + 8}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.3.3

Reorder the factors of $(t + h + 8) (t + 8)$ .

$\frac{\frac{t + 8}{(t + 8) (t + h + 8)} - \frac{t + h + 8}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{t + 8 - (t + h + 8)}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.5

Rewrite $\frac{t + 8 - (t + h + 8)}{(t + 8) (t + h + 8)}$ in a factored form.

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Step 4.1.5.1

Apply the distributive property.

$\frac{\frac{t + 8 - t - h - 1 \cdot 8}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.5.2

Multiply $- 1$ by $8$ .

$\frac{\frac{t + 8 - t - h - 8}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.5.3

Subtract $t$ from $t$ .

$\frac{\frac{0 + 8 - h - 8}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.5.4

Add $0$ and $8$ .

$\frac{\frac{8 - h - 8}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.5.5

Subtract $8$ from $8$ .

$\frac{\frac{- h + 0}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.5.6

Add $- h$ and $0$ .

$\frac{\frac{- h}{(t + 8) (t + h + 8)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{h}{(t + 8) (t + h + 8)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{h}{(t + 8) (t + h + 8)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{h}{(t + 8) (t + h + 8)}$ into the numerator.

$\frac{- h}{(t + 8) (t + h + 8)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- h$ .

$\frac{h \cdot - 1}{(t + 8) (t + h + 8)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 1}{(t + 8) (t + h + 8)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 1}{(t + 8) (t + h + 8)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{1}{(t + 8) (t + h + 8)}$

Step 5