Precalculus Examples

$f (t) = \frac{95 t}{99 t - 95}$

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.

$f (t + h) = \frac{95 (t + h)}{99 (t + h) - 95}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (t + h) = \frac{95 (t + h)}{99 t + 99 h - 95}$

Step 2.1.2.2

The final answer is $\frac{95 (t + h)}{99 t + 99 h - 95}$ .

$\frac{95 (t + h)}{99 t + 99 h - 95}$

Step 2.2

Find the components of the definition.

$f (t + h) = \frac{95 (t + h)}{99 t + 99 h - 95}$

$f (t) = \frac{95 t}{99 t - 95}$

$f (t + h) = \frac{95 (t + h)}{99 t + 99 h - 95}$

$f (t) = \frac{95 t}{99 t - 95}$

Step 3

Plug in the components.

$\frac{f (t + h) - f t}{h} = \frac{\frac{95 (t + h)}{99 t + 99 h - 95} - (\frac{95 t}{99 t - 95})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{95 (t + h)}{99 t + 99 h - 95}$ as a fraction with a common denominator, multiply by $\frac{99 t - 95}{99 t - 95}$ .

$\frac{\frac{95 (t + h)}{99 t + 99 h - 95} \cdot \frac{99 t - 95}{99 t - 95} - \frac{95 t}{99 t - 95}}{h}$

Step 4.1.2

To write $- \frac{95 t}{99 t - 95}$ as a fraction with a common denominator, multiply by $\frac{99 t + 99 h - 95}{99 t + 99 h - 95}$ .

$\frac{\frac{95 (t + h)}{99 t + 99 h - 95} \cdot \frac{99 t - 95}{99 t - 95} - \frac{95 t}{99 t - 95} \cdot \frac{99 t + 99 h - 95}{99 t + 99 h - 95}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(99 t + 99 h - 95) (99 t - 95)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{95 (t + h)}{99 t + 99 h - 95}$ by $\frac{99 t - 95}{99 t - 95}$ .

$\frac{\frac{95 (t + h) (99 t - 95)}{(99 t + 99 h - 95) (99 t - 95)} - \frac{95 t}{99 t - 95} \cdot \frac{99 t + 99 h - 95}{99 t + 99 h - 95}}{h}$

Step 4.1.3.2

Multiply $\frac{95 t}{99 t - 95}$ by $\frac{99 t + 99 h - 95}{99 t + 99 h - 95}$ .

$\frac{\frac{95 (t + h) (99 t - 95)}{(99 t + 99 h - 95) (99 t - 95)} - \frac{95 t (99 t + 99 h - 95)}{(99 t - 95) (99 t + 99 h - 95)}}{h}$

Step 4.1.3.3

Reorder the factors of $(99 t - 95) (99 t + 99 h - 95)$ .

$\frac{\frac{95 (t + h) (99 t - 95)}{(99 t + 99 h - 95) (99 t - 95)} - \frac{95 t (99 t + 99 h - 95)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{95 (t + h) (99 t - 95) - 95 t (99 t + 99 h - 95)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5

Rewrite $\frac{95 (t + h) (99 t - 95) - 95 t (99 t + 99 h - 95)}{(99 t + 99 h - 95) (99 t - 95)}$ in a factored form.

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Step 4.1.5.1

Factor $95$ out of $95 (t + h) (99 t - 95) - 95 t (99 t + 99 h - 95)$ .

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Step 4.1.5.1.1

Factor $95$ out of $95 (t + h) (99 t - 95)$ .

$\frac{\frac{95 ((t + h) (99 t - 95)) - 95 t (99 t + 99 h - 95)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.1.2

Factor $95$ out of $- 95 t (99 t + 99 h - 95)$ .

$\frac{\frac{95 ((t + h) (99 t - 95)) + 95 (- t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.1.3

Factor $95$ out of $95 ((t + h) (99 t - 95)) + 95 (- t (99 t + 99 h - 95))$ .

$\frac{\frac{95 ((t + h) (99 t - 95) - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.2

Expand $(t + h) (99 t - 95)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

$\frac{\frac{95 (t (99 t - 95) + h (99 t - 95) - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.2.2

Apply the distributive property.

$\frac{\frac{95 (t (99 t) + t \cdot - 95 + h (99 t - 95) - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.2.3

Apply the distributive property.

$\frac{\frac{95 (t (99 t) + t \cdot - 95 + h (99 t) + h \cdot - 95 - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

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.

$\frac{\frac{95 (99 t \cdot t + t \cdot - 95 + h (99 t) + h \cdot - 95 - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

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 $t$ .

$\frac{\frac{95 (99 (t \cdot t) + t \cdot - 95 + h (99 t) + h \cdot - 95 - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.3.2.2

Multiply $t$ by $t$ .

$\frac{\frac{95 (99 t^{2} + t \cdot - 95 + h (99 t) + h \cdot - 95 - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.3.3

Move $- 95$ to the left of $t$ .

$\frac{\frac{95 (99 t^{2} - 95 \cdot t + h (99 t) + h \cdot - 95 - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.3.4

Rewrite using the commutative property of multiplication.

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t + h \cdot - 95 - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.3.5

Move $- 95$ to the left of $h$ .

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - t (99 t + 99 h - 95))}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.4

Apply the distributive property.

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - t (99 t) - t (99 h) - t \cdot - 95)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.5

Simplify.

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Step 4.1.5.5.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - 1 \cdot 99 t \cdot t - t (99 h) - t \cdot - 95)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.5.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - 1 \cdot 99 t \cdot t - 1 \cdot 99 t h - t \cdot - 95)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.5.3

Multiply $- 95$ by $- 1$ .

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - 1 \cdot 99 t \cdot t - 1 \cdot 99 t h + 95 t)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

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 $t$ .

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - 1 \cdot 99 (t \cdot t) - 1 \cdot 99 t h + 95 t)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.6.1.2

Multiply $t$ by $t$ .

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - 1 \cdot 99 t^{2} - 1 \cdot 99 t h + 95 t)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.6.2

Multiply $- 1$ by $99$ .

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - 99 t^{2} - 1 \cdot 99 t h + 95 t)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.6.3

Multiply $- 1$ by $99$ .

$\frac{\frac{95 (99 t^{2} - 95 t + 99 h t - 95 h - 99 t^{2} - 99 t h + 95 t)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.7

Subtract $99 t^{2}$ from $99 t^{2}$ .

$\frac{\frac{95 (- 95 t + 99 h t - 95 h + 0 - 99 t h + 95 t)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.8

Add $- 95 t$ and $0$ .

$\frac{\frac{95 (- 95 t + 99 h t - 95 h - 99 t h + 95 t)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.9

Add $- 95 t$ and $95 t$ .

$\frac{\frac{95 (99 h t - 95 h - 99 t h + 0)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.10

Add $99 h t - 95 h - 99 t h$ and $0$ .

$\frac{\frac{95 (99 h t - 95 h - 99 t h)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.11

Subtract $99 t h$ from $99 h t$ .

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Step 4.1.5.11.1

Move $h$ .

$\frac{\frac{95 (- 95 h + 99 t h - 99 t h)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.11.2

Subtract $99 t h$ from $99 t h$ .

$\frac{\frac{95 (- 95 h + 0)}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.12

Add $- 95 h$ and $0$ .

$\frac{\frac{95 \cdot - 95 h}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.5.13

Multiply $95$ by $- 95$ .

$\frac{\frac{- 9025 h}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{9025 h}{(99 t + 99 h - 95) (99 t - 95)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{9025 h}{(99 t + 99 h - 95) (99 t - 95)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{9025 h}{(99 t + 99 h - 95) (99 t - 95)}$ into the numerator.

$\frac{- 9025 h}{(99 t + 99 h - 95) (99 t - 95)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 9025 h$ .

$\frac{h \cdot - 9025}{(99 t + 99 h - 95) (99 t - 95)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 9025}{(99 t + 99 h - 95) (99 t - 95)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 9025}{(99 t + 99 h - 95) (99 t - 95)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{9025}{(99 t + 99 h - 95) (99 t - 95)}$

Step 5