Precalculus Examples

$d (t) = \frac{100 t}{2 t + 15}$

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.

$d (t + h) = \frac{100 (t + h)}{2 (t + h) + 15}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$d (t + h) = \frac{100 (t + h)}{2 t + 2 h + 15}$

Step 2.1.2.2

The final answer is $\frac{100 (t + h)}{2 t + 2 h + 15}$ .

$\frac{100 (t + h)}{2 t + 2 h + 15}$

Step 2.2

Find the components of the definition.

$d (t + h) = \frac{100 (t + h)}{2 t + 2 h + 15}$

$d (t) = \frac{100 t}{2 t + 15}$

$d (t + h) = \frac{100 (t + h)}{2 t + 2 h + 15}$

$d (t) = \frac{100 t}{2 t + 15}$

Step 3

Plug in the components.

$\frac{d (t + h) - d t}{h} = \frac{\frac{100 (t + h)}{2 t + 2 h + 15} - (\frac{100 t}{2 t + 15})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{100 (t + h)}{2 t + 2 h + 15}$ as a fraction with a common denominator, multiply by $\frac{2 t + 15}{2 t + 15}$ .

$\frac{\frac{100 (t + h)}{2 t + 2 h + 15} \cdot \frac{2 t + 15}{2 t + 15} - \frac{100 t}{2 t + 15}}{h}$

Step 4.1.2

To write $- \frac{100 t}{2 t + 15}$ as a fraction with a common denominator, multiply by $\frac{2 t + 2 h + 15}{2 t + 2 h + 15}$ .

$\frac{\frac{100 (t + h)}{2 t + 2 h + 15} \cdot \frac{2 t + 15}{2 t + 15} - \frac{100 t}{2 t + 15} \cdot \frac{2 t + 2 h + 15}{2 t + 2 h + 15}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(2 t + 2 h + 15) (2 t + 15)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{100 (t + h)}{2 t + 2 h + 15}$ by $\frac{2 t + 15}{2 t + 15}$ .

$\frac{\frac{100 (t + h) (2 t + 15)}{(2 t + 2 h + 15) (2 t + 15)} - \frac{100 t}{2 t + 15} \cdot \frac{2 t + 2 h + 15}{2 t + 2 h + 15}}{h}$

Step 4.1.3.2

Multiply $\frac{100 t}{2 t + 15}$ by $\frac{2 t + 2 h + 15}{2 t + 2 h + 15}$ .

$\frac{\frac{100 (t + h) (2 t + 15)}{(2 t + 2 h + 15) (2 t + 15)} - \frac{100 t (2 t + 2 h + 15)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.3.3

Reorder the factors of $(2 t + 2 h + 15) (2 t + 15)$ .

$\frac{\frac{100 (t + h) (2 t + 15)}{(2 t + 15) (2 t + 2 h + 15)} - \frac{100 t (2 t + 2 h + 15)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{100 (t + h) (2 t + 15) - 100 t (2 t + 2 h + 15)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5

Rewrite $\frac{100 (t + h) (2 t + 15) - 100 t (2 t + 2 h + 15)}{(2 t + 15) (2 t + 2 h + 15)}$ in a factored form.

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Step 4.1.5.1

Factor $100$ out of $100 (t + h) (2 t + 15) - 100 t (2 t + 2 h + 15)$ .

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Step 4.1.5.1.1

Factor $100$ out of $100 (t + h) (2 t + 15)$ .

$\frac{\frac{100 ((t + h) (2 t + 15)) - 100 t (2 t + 2 h + 15)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.1.2

Factor $100$ out of $- 100 t (2 t + 2 h + 15)$ .

$\frac{\frac{100 ((t + h) (2 t + 15)) + 100 (- t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.1.3

Factor $100$ out of $100 ((t + h) (2 t + 15)) + 100 (- t (2 t + 2 h + 15))$ .

$\frac{\frac{100 ((t + h) (2 t + 15) - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.2

Expand $(t + h) (2 t + 15)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

$\frac{\frac{100 (t (2 t + 15) + h (2 t + 15) - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.2.2

Apply the distributive property.

$\frac{\frac{100 (t (2 t) + t \cdot 15 + h (2 t + 15) - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.2.3

Apply the distributive property.

$\frac{\frac{100 (t (2 t) + t \cdot 15 + h (2 t) + h \cdot 15 - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.3

Simplify each term.

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Step 4.1.5.3.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{100 (2 t \cdot t + t \cdot 15 + h (2 t) + h \cdot 15 - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.3.2

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

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Step 4.1.5.3.2.1

Move $t$ .

$\frac{\frac{100 (2 (t \cdot t) + t \cdot 15 + h (2 t) + h \cdot 15 - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.3.2.2

Multiply $t$ by $t$ .

$\frac{\frac{100 (2 t^{2} + t \cdot 15 + h (2 t) + h \cdot 15 - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.3.3

Move $15$ to the left of $t$ .

$\frac{\frac{100 (2 t^{2} + 15 \cdot t + h (2 t) + h \cdot 15 - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.3.4

Rewrite using the commutative property of multiplication.

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + h \cdot 15 - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.3.5

Move $15$ to the left of $h$ .

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - t (2 t + 2 h + 15))}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.4

Apply the distributive property.

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - t (2 t) - t (2 h) - t \cdot 15)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.5

Simplify.

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Step 4.1.5.5.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - 1 \cdot 2 t \cdot t - t (2 h) - t \cdot 15)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.5.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - 1 \cdot 2 t \cdot t - 1 \cdot 2 t h - t \cdot 15)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.5.3

Multiply $15$ by $- 1$ .

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - 1 \cdot 2 t \cdot t - 1 \cdot 2 t h - 15 t)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.6

Simplify each term.

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Step 4.1.5.6.1

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

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Step 4.1.5.6.1.1

Move $t$ .

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - 1 \cdot 2 (t \cdot t) - 1 \cdot 2 t h - 15 t)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.6.1.2

Multiply $t$ by $t$ .

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - 1 \cdot 2 t^{2} - 1 \cdot 2 t h - 15 t)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.6.2

Multiply $- 1$ by $2$ .

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - 2 t^{2} - 1 \cdot 2 t h - 15 t)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.6.3

Multiply $- 1$ by $2$ .

$\frac{\frac{100 (2 t^{2} + 15 t + 2 h t + 15 h - 2 t^{2} - 2 t h - 15 t)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.7

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

$\frac{\frac{100 (15 t + 2 h t + 15 h + 0 - 2 t h - 15 t)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.8

Add $15 t$ and $0$ .

$\frac{\frac{100 (15 t + 2 h t + 15 h - 2 t h - 15 t)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.9

Subtract $15 t$ from $15 t$ .

$\frac{\frac{100 (2 h t + 15 h - 2 t h + 0)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.10

Add $2 h t + 15 h - 2 t h$ and $0$ .

$\frac{\frac{100 (2 h t + 15 h - 2 t h)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.11

Subtract $2 t h$ from $2 h t$ .

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Step 4.1.5.11.1

Move $h$ .

$\frac{\frac{100 (15 h + 2 t h - 2 t h)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.11.2

Subtract $2 t h$ from $2 t h$ .

$\frac{\frac{100 (15 h + 0)}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.12

Add $15 h$ and $0$ .

$\frac{\frac{100 \cdot 15 h}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.1.5.13

Multiply $100$ by $15$ .

$\frac{\frac{1500 h}{(2 t + 15) (2 t + 2 h + 15)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1500 h}{(2 t + 15) (2 t + 2 h + 15)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $1500 h$ .

$\frac{h \cdot 1500}{(2 t + 15) (2 t + 2 h + 15)} \cdot \frac{1}{h}$

Step 4.3.2

Cancel the common factor.

$\frac{h \cdot 1500}{(2 t + 15) (2 t + 2 h + 15)} \cdot \frac{1}{h}$

Step 4.3.3

Rewrite the expression.

$\frac{1500}{(2 t + 15) (2 t + 2 h + 15)}$

Step 5