Precalculus Examples

$S (t) = 155 (7 - \frac{7}{6 + t})$

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.

$S (t + h) = 155 (7 - \frac{7}{6 + t + h})$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$S (t + h) = 155 (7 - \frac{7}{6 + t + h})$

Step 2.1.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{6 + t + h}{6 + t + h}$ .

$S (t + h) = 155 (\frac{7 (6 + t + h)}{6 + t + h} - \frac{7}{6 + t + h})$

Step 2.1.2.3

Combine the numerators over the common denominator.

$S (t + h) = 155 (\frac{7 (6 + t + h) - 7}{6 + t + h})$

Step 2.1.2.4

Simplify the numerator.

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Step 2.1.2.4.1

Factor $7$ out of $7 (6 + t + h) - 7$ .

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Step 2.1.2.4.1.1

Factor $7$ out of $- 7$ .

$S (t + h) = 155 (\frac{7 (6 + t + h) + 7 (- 1)}{6 + t + h})$

Step 2.1.2.4.1.2

Factor $7$ out of $7 (6 + t + h) + 7 (- 1)$ .

$S (t + h) = 155 (\frac{7 (6 + t + h - 1)}{6 + t + h})$

Step 2.1.2.4.2

Subtract $1$ from $6$ .

$S (t + h) = 155 (\frac{7 (t + h + 5)}{6 + t + h})$

Step 2.1.2.5

Multiply $155 \frac{7 (t + h + 5)}{6 + t + h}$ .

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Step 2.1.2.5.1

Combine $155$ and $\frac{7 (t + h + 5)}{6 + t + h}$ .

$S (t + h) = \frac{155 (7 (t + h + 5))}{6 + t + h}$

Step 2.1.2.5.2

Multiply $7$ by $155$ .

$S (t + h) = \frac{1085 (t + h + 5)}{6 + t + h}$

Step 2.1.2.6

The final answer is $\frac{1085 (t + h + 5)}{6 + t + h}$ .

$\frac{1085 (t + h + 5)}{6 + t + h}$

Step 2.2

Find the components of the definition.

$S (t + h) = \frac{1085 (t + h + 5)}{6 + t + h}$

$S (t) = \frac{1085 (t + 5)}{6 + t}$

$S (t + h) = \frac{1085 (t + h + 5)}{6 + t + h}$

$S (t) = \frac{1085 (t + 5)}{6 + t}$

Step 3

Plug in the components.

$\frac{S (t + h) - S t}{h} = \frac{\frac{1085 (t + h + 5)}{6 + t + h} - (\frac{1085 (t + 5)}{6 + t})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1085 (t + h + 5)}{6 + t + h}$ as a fraction with a common denominator, multiply by $\frac{6 + t}{6 + t}$ .

$\frac{\frac{1085 (t + h + 5)}{6 + t + h} \cdot \frac{6 + t}{6 + t} - \frac{1085 (t + 5)}{6 + t}}{h}$

Step 4.1.2

To write $- \frac{1085 (t + 5)}{6 + t}$ as a fraction with a common denominator, multiply by $\frac{6 + t + h}{6 + t + h}$ .

$\frac{\frac{1085 (t + h + 5)}{6 + t + h} \cdot \frac{6 + t}{6 + t} - \frac{1085 (t + 5)}{6 + t} \cdot \frac{6 + t + h}{6 + t + h}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(6 + t + h) (6 + t)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{1085 (t + h + 5)}{6 + t + h}$ by $\frac{6 + t}{6 + t}$ .

$\frac{\frac{1085 (t + h + 5) (6 + t)}{(6 + t + h) (6 + t)} - \frac{1085 (t + 5)}{6 + t} \cdot \frac{6 + t + h}{6 + t + h}}{h}$

Step 4.1.3.2

Multiply $\frac{1085 (t + 5)}{6 + t}$ by $\frac{6 + t + h}{6 + t + h}$ .

$\frac{\frac{1085 (t + h + 5) (6 + t)}{(6 + t + h) (6 + t)} - \frac{1085 (t + 5) (6 + t + h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.3.3

Reorder the factors of $(6 + t + h) (6 + t)$ .

$\frac{\frac{1085 (t + h + 5) (6 + t)}{(6 + t) (6 + t + h)} - \frac{1085 (t + 5) (6 + t + h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{1085 (t + h + 5) (6 + t) - 1085 (t + 5) (6 + t + h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5

Rewrite $\frac{1085 (t + h + 5) (6 + t) - 1085 (t + 5) (6 + t + h)}{(6 + t) (6 + t + h)}$ in a factored form.

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Step 4.1.5.1

Factor $1085$ out of $1085 (t + h + 5) (6 + t) - 1085 (t + 5) (6 + t + h)$ .

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Step 4.1.5.1.1

Factor $1085$ out of $1085 (t + h + 5) (6 + t)$ .

$\frac{\frac{1085 ((t + h + 5) (6 + t)) - 1085 (t + 5) (6 + t + h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.1.2

Factor $1085$ out of $- 1085 (t + 5) (6 + t + h)$ .

$\frac{\frac{1085 ((t + h + 5) (6 + t)) + 1085 (- (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.1.3

Factor $1085$ out of $1085 ((t + h + 5) (6 + t)) + 1085 (- (t + 5) (6 + t + h))$ .

$\frac{\frac{1085 ((t + h + 5) (6 + t) - (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.2

Expand $(t + h + 5) (6 + t)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{1085 (t \cdot 6 + t \cdot t + h \cdot 6 + h t + 5 \cdot 6 + 5 t - (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.3

Simplify each term.

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Step 4.1.5.3.1

Move $6$ to the left of $t$ .

$\frac{\frac{1085 (6 \cdot t + t \cdot t + h \cdot 6 + h t + 5 \cdot 6 + 5 t - (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.3.2

Multiply $t$ by $t$ .

$\frac{\frac{1085 (6 t + t^{2} + h \cdot 6 + h t + 5 \cdot 6 + 5 t - (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.3.3

Move $6$ to the left of $h$ .

$\frac{\frac{1085 (6 t + t^{2} + 6 \cdot h + h t + 5 \cdot 6 + 5 t - (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.3.4

Multiply $5$ by $6$ .

$\frac{\frac{1085 (6 t + t^{2} + 6 h + h t + 30 + 5 t - (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.4

Add $6 t$ and $5 t$ .

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 - (t + 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.5

Apply the distributive property.

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 + (- t - 1 \cdot 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.6

Multiply $- 1$ by $5$ .

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 + (- t - 5) (6 + t + h))}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.7

Expand $(- t - 5) (6 + t + h)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 - t \cdot 6 - t \cdot t - t h - 5 \cdot 6 - 5 t - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.8

Simplify each term.

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Step 4.1.5.8.1

Multiply $6$ by $- 1$ .

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 - 6 t - t \cdot t - t h - 5 \cdot 6 - 5 t - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.8.2

Multiply $t$ by $t$ by adding the exponents.

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Step 4.1.5.8.2.1

Move $t$ .

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 - 6 t - (t \cdot t) - t h - 5 \cdot 6 - 5 t - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.8.2.2

Multiply $t$ by $t$ .

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 - 6 t - t^{2} - t h - 5 \cdot 6 - 5 t - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.8.3

Multiply $- 5$ by $6$ .

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 - 6 t - t^{2} - t h - 30 - 5 t - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.9

Subtract $5 t$ from $- 6 t$ .

$\frac{\frac{1085 (11 t + t^{2} + 6 h + h t + 30 - t^{2} - t h - 30 - 11 t - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.10

Subtract $11 t$ from $11 t$ .

$\frac{\frac{1085 (t^{2} + 6 h + h t + 30 - t^{2} - t h - 30 + 0 - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.11

Subtract $t^{2}$ from $t^{2}$ .

$\frac{\frac{1085 (6 h + h t + 30 + 0 - t h - 30 + 0 - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.12

Add $6 h$ and $0$ .

$\frac{\frac{1085 (6 h + h t + 30 - t h - 30 + 0 - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.13

Add $6 h$ and $0$ .

$\frac{\frac{1085 (6 h + h t + 30 - t h - 30 - 5 h)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.14

Subtract $5 h$ from $6 h$ .

$\frac{\frac{1085 (h + h t + 30 - t h - 30)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.15

Subtract $t h$ from $h t$ .

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Step 4.1.5.15.1

Reorder $h$ and $t$ .

$\frac{\frac{1085 (h + t h - t h + 30 - 30)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.15.2

Subtract $t h$ from $t h$ .

$\frac{\frac{1085 (h + 0 + 30 - 30)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.16

Add $h$ and $0$ .

$\frac{\frac{1085 (h + 30 - 30)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.17

Subtract $30$ from $30$ .

$\frac{\frac{1085 (h + 0)}{(6 + t) (6 + t + h)}}{h}$

Step 4.1.5.18

Add $h$ and $0$ .

$\frac{\frac{1085 h}{(6 + t) (6 + t + h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1085 h}{(6 + t) (6 + t + h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $1085 h$ .

$\frac{h \cdot 1085}{(6 + t) (6 + t + h)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$\frac{h \cdot 1085}{(6 + t) (6 + t + h)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$\frac{1085}{(6 + t) (6 + t + h)}$

Step 5