Precalculus Examples

$C (t) = \frac{0.2 t}{t^{2} + 4}$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$C (t + h) = \frac{0.2 (t + h)}{{(t + h)}^{2} + 4}$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Simplify the denominator.

Tap for more steps...

Step 2.1.2.1.1

Rewrite ${(t + h)}^{2}$ as $(t + h) (t + h)$ .

$C (t + h) = \frac{0.2 (t + h)}{(t + h) (t + h) + 4}$

Step 2.1.2.1.2

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

Tap for more steps...

Step 2.1.2.1.2.1

Apply the distributive property.

$C (t + h) = \frac{0.2 (t + h)}{t (t + h) + h (t + h) + 4}$

Step 2.1.2.1.2.2

Apply the distributive property.

$C (t + h) = \frac{0.2 (t + h)}{t \cdot t + t h + h (t + h) + 4}$

Step 2.1.2.1.2.3

Apply the distributive property.

$C (t + h) = \frac{0.2 (t + h)}{t \cdot t + t h + h t + h \cdot h + 4}$

Step 2.1.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.3.1.1

Multiply $t$ by $t$ .

$C (t + h) = \frac{0.2 (t + h)}{t^{2} + t h + h t + h \cdot h + 4}$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$C (t + h) = \frac{0.2 (t + h)}{t^{2} + t h + h t + h^{2} + 4}$

Step 2.1.2.1.3.2

Add $t h$ and $h t$ .

Tap for more steps...

Step 2.1.2.1.3.2.1

Reorder $t$ and $h$ .

$C (t + h) = \frac{0.2 (t + h)}{t^{2} + h t + h t + h^{2} + 4}$

Step 2.1.2.1.3.2.2

Add $h t$ and $h t$ .

$C (t + h) = \frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4}$

Step 2.1.2.2

The final answer is $\frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4}$ .

$\frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4}$

Step 2.2

Find the components of the definition.

$C (t + h) = \frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4}$

$C (t) = \frac{0.2 t}{t^{2} + 4}$

$C (t + h) = \frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4}$

$C (t) = \frac{0.2 t}{t^{2} + 4}$

Step 3

Plug in the components.

$\frac{C (t + h) - C t}{h} = \frac{\frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4} - (\frac{0.2 t}{t^{2} + 4})}{h}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

To write $\frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4}$ as a fraction with a common denominator, multiply by $\frac{t^{2} + 4}{t^{2} + 4}$ .

$\frac{\frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4} \cdot \frac{t^{2} + 4}{t^{2} + 4} - \frac{0.2 t}{t^{2} + 4}}{h}$

Step 4.1.2

To write $- \frac{0.2 t}{t^{2} + 4}$ as a fraction with a common denominator, multiply by $\frac{t^{2} + 2 h t + h^{2} + 4}{t^{2} + 2 h t + h^{2} + 4}$ .

$\frac{\frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4} \cdot \frac{t^{2} + 4}{t^{2} + 4} - \frac{0.2 t}{t^{2} + 4} \cdot \frac{t^{2} + 2 h t + h^{2} + 4}{t^{2} + 2 h t + h^{2} + 4}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1.3.1

Multiply $\frac{0.2 (t + h)}{t^{2} + 2 h t + h^{2} + 4}$ by $\frac{t^{2} + 4}{t^{2} + 4}$ .

$\frac{\frac{0.2 (t + h) (t^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)} - \frac{0.2 t}{t^{2} + 4} \cdot \frac{t^{2} + 2 h t + h^{2} + 4}{t^{2} + 2 h t + h^{2} + 4}}{h}$

Step 4.1.3.2

Multiply $\frac{0.2 t}{t^{2} + 4}$ by $\frac{t^{2} + 2 h t + h^{2} + 4}{t^{2} + 2 h t + h^{2} + 4}$ .

$\frac{\frac{0.2 (t + h) (t^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)} - \frac{0.2 t (t^{2} + 2 h t + h^{2} + 4)}{(t^{2} + 4) (t^{2} + 2 h t + h^{2} + 4)}}{h}$

Step 4.1.3.3

Reorder the factors of $(t^{2} + 4) (t^{2} + 2 h t + h^{2} + 4)$ .

$\frac{\frac{0.2 (t + h) (t^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)} - \frac{0.2 t (t^{2} + 2 h t + h^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{0.2 (t + h) (t^{2} + 4) - 0.2 t (t^{2} + 2 h t + h^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5

Rewrite $\frac{0.2 (t + h) (t^{2} + 4) - 0.2 t (t^{2} + 2 h t + h^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$ in a factored form.

Tap for more steps...

Step 4.1.5.1

Factor $0.2$ out of $0.2 (t + h) (t^{2} + 4) - 0.2 t (t^{2} + 2 h t + h^{2} + 4)$ .

Tap for more steps...

Step 4.1.5.1.1

Move $h$ .

$\frac{\frac{0.2 (t + h) (t^{2} + 4) - 0.2 t (t^{2} + 2 t h + h^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.1.2

Factor $0.2$ out of $0.2 (t + h) (t^{2} + 4)$ .

$\frac{\frac{0.2 ((t + h) (t^{2} + 4)) - 0.2 t (t^{2} + 2 t h + h^{2} + 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.1.3

Factor $0.2$ out of $- 0.2 t (t^{2} + 2 t h + h^{2} + 4)$ .

$\frac{\frac{0.2 ((t + h) (t^{2} + 4)) + 0.2 (- t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.1.4

Factor $0.2$ out of $0.2 ((t + h) (t^{2} + 4)) + 0.2 (- t (t^{2} + 2 t h + h^{2} + 4))$ .

$\frac{\frac{0.2 ((t + h) (t^{2} + 4) - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.2

Expand $(t + h) (t^{2} + 4)$ using the FOIL Method.

Tap for more steps...

Step 4.1.5.2.1

Apply the distributive property.

$\frac{\frac{0.2 (t (t^{2} + 4) + h (t^{2} + 4) - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.2.2

Apply the distributive property.

$\frac{\frac{0.2 (t \cdot t^{2} + t \cdot 4 + h (t^{2} + 4) - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.2.3

Apply the distributive property.

$\frac{\frac{0.2 (t \cdot t^{2} + t \cdot 4 + h t^{2} + h \cdot 4 - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.3

Simplify each term.

Tap for more steps...

Step 4.1.5.3.1

Multiply $t$ by $t^{2}$ by adding the exponents.

Tap for more steps...

Step 4.1.5.3.1.1

Multiply $t$ by $t^{2}$ .

Tap for more steps...

Step 4.1.5.3.1.1.1

Raise $t$ to the power of $1$ .

$\frac{\frac{0.2 (t^{1} t^{2} + t \cdot 4 + h t^{2} + h \cdot 4 - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{0.2 (t^{1 + 2} + t \cdot 4 + h t^{2} + h \cdot 4 - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.3.1.2

Add $1$ and $2$ .

$\frac{\frac{0.2 (t^{3} + t \cdot 4 + h t^{2} + h \cdot 4 - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.3.2

Move $4$ to the left of $t$ .

$\frac{\frac{0.2 (t^{3} + 4 \cdot t + h t^{2} + h \cdot 4 - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.3.3

Move $4$ to the left of $h$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t (t^{2} + 2 t h + h^{2} + 4))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.4

Apply the distributive property.

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t \cdot t^{2} - t (2 t h) - t h^{2} - t \cdot 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.5

Simplify.

Tap for more steps...

Step 4.1.5.5.1

Multiply $t$ by $t^{2}$ by adding the exponents.

Tap for more steps...

Step 4.1.5.5.1.1

Move $t^{2}$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - (t^{2} t) - t (2 t h) - t h^{2} - t \cdot 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.5.1.2

Multiply $t^{2}$ by $t$ .

Tap for more steps...

Step 4.1.5.5.1.2.1

Raise $t$ to the power of $1$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - (t^{2} t^{1}) - t (2 t h) - t h^{2} - t \cdot 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.5.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t^{2 + 1} - t (2 t h) - t h^{2} - t \cdot 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.5.1.3

Add $2$ and $1$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t^{3} - t (2 t h) - t h^{2} - t \cdot 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.5.2

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

Tap for more steps...

Step 4.1.5.5.2.1

Move $t$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t^{3} - (t \cdot t) (2 h) - t h^{2} - t \cdot 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.5.2.2

Multiply $t$ by $t$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t^{3} - t^{2} (2 h) - t h^{2} - t \cdot 4)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.5.3

Multiply $4$ by $- 1$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t^{3} - t^{2} (2 h) - t h^{2} - 4 t)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.6

Simplify each term.

Tap for more steps...

Step 4.1.5.6.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t^{3} - 1 \cdot 2 t^{2} h - t h^{2} - 4 t)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.6.2

Multiply $- 1$ by $2$ .

$\frac{\frac{0.2 (t^{3} + 4 t + h t^{2} + 4 h - t^{3} - 2 t^{2} h - t h^{2} - 4 t)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.7

Subtract $t^{3}$ from $t^{3}$ .

$\frac{\frac{0.2 (4 t + h t^{2} + 4 h + 0 - 2 t^{2} h - t h^{2} - 4 t)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.8

Add $4 t$ and $0$ .

$\frac{\frac{0.2 (4 t + h t^{2} + 4 h - 2 t^{2} h - t h^{2} - 4 t)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.9

Subtract $4 t$ from $4 t$ .

$\frac{\frac{0.2 (h t^{2} + 4 h - 2 t^{2} h - t h^{2} + 0)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.10

Add $h t^{2} + 4 h - 2 t^{2} h - t h^{2}$ and $0$ .

$\frac{\frac{0.2 (h t^{2} + 4 h - 2 t^{2} h - t h^{2})}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.11

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

Tap for more steps...

Step 4.1.5.11.1

Reorder $h$ and $t^{2}$ .

$\frac{\frac{0.2 (t^{2} h - 2 t^{2} h + 4 h - t h^{2})}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.11.2

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

$\frac{\frac{0.2 (- t^{2} h + 4 h - t h^{2})}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.12

Factor $h$ out of $- t^{2} h + 4 h - t h^{2}$ .

Tap for more steps...

Step 4.1.5.12.1

Factor $h$ out of $- t^{2} h$ .

$\frac{\frac{0.2 (h (- t^{2}) + 4 h - t h^{2})}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.12.2

Factor $h$ out of $4 h$ .

$\frac{\frac{0.2 (h (- t^{2}) + h \cdot 4 - t h^{2})}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.12.3

Factor $h$ out of $- t h^{2}$ .

$\frac{\frac{0.2 (h (- t^{2}) + h \cdot 4 + h (- t h))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.12.4

Factor $h$ out of $h (- t^{2}) + h \cdot 4$ .

$\frac{\frac{0.2 (h (- t^{2} + 4) + h (- t h))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.1.5.12.5

Factor $h$ out of $h (- t^{2} + 4) + h (- t h)$ .

$\frac{\frac{0.2 (h (- t^{2} + 4 - t h))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

$\frac{\frac{0.2 h (- t^{2} + 4 - t h)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.2 h (- t^{2} + 4 - t h)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)} \cdot \frac{1}{h}$

Step 4.3

Combine.

$\frac{0.2 h (- t^{2} + 4 - t h) \cdot 1}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4) h}$

Step 4.4

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.4.1

Cancel the common factor.

$\frac{0.2 h (- t^{2} + 4 - t h) \cdot 1}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4) h}$

Step 4.4.2

Rewrite the expression.

$\frac{0.2 (- t^{2} + 4 - t h) \cdot 1}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.5

Multiply $0.2$ by $1$ .

$\frac{0.2 (- t^{2} + 4 - t h)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.6

Factor $- 1$ out of $- t^{2}$ .

$\frac{0.2 (- (t^{2}) + 4 - t h)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.7

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{0.2 (- (t^{2}) - 1 (- 4) - t h)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.8

Factor $- 1$ out of $- (t^{2}) - 1 (- 4)$ .

$\frac{0.2 (- (t^{2} - 4) - t h)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.9

Factor $- 1$ out of $- t h$ .

$\frac{0.2 (- (t^{2} - 4) - (t h))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.10

Factor $- 1$ out of $- (t^{2} - 4) - (t h)$ .

$\frac{0.2 (- (t^{2} - 4 + t h))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.11

Simplify the expression.

Tap for more steps...

Step 4.11.1

Rewrite $- (t^{2} - 4 + t h)$ as $- 1 (t^{2} - 4 + t h)$ .

$\frac{0.2 (- 1 (t^{2} - 4 + t h))}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 4.11.2

Move the negative in front of the fraction.

$- \frac{0.2 (t^{2} - 4 + t h)}{(t^{2} + 2 h t + h^{2} + 4) (t^{2} + 4)}$

Step 5