Precalculus Examples

$h (t) = {(t + 3)}^{2} + 5$

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.

$h (t + h) = {((t + h) + 3)}^{2} + 5$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Rewrite ${(t + h + 3)}^{2}$ as $(t + h + 3) (t + h + 3)$ .

$h (t + h) = (t + h + 3) (t + h + 3) + 5$

Step 2.1.2.1.2

Expand $(t + h + 3) (t + h + 3)$ by multiplying each term in the first expression by each term in the second expression.

$h (t + h) = t \cdot t + t h + t \cdot 3 + h t + h \cdot h + h \cdot 3 + 3 t + 3 h + 3 \cdot 3 + 5$

Step 2.1.2.1.3

Simplify each term.

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Step 2.1.2.1.3.1

Multiply $t$ by $t$ .

$h (t + h) = t^{2} + t h + t \cdot 3 + h t + h \cdot h + h \cdot 3 + 3 t + 3 h + 3 \cdot 3 + 5$

Step 2.1.2.1.3.2

Move $3$ to the left of $t$ .

$h (t + h) = t^{2} + t h + 3 \cdot t + h t + h \cdot h + h \cdot 3 + 3 t + 3 h + 3 \cdot 3 + 5$

Step 2.1.2.1.3.3

Multiply $h$ by $h$ .

$h (t + h) = t^{2} + t h + 3 t + h t + h^{2} + h \cdot 3 + 3 t + 3 h + 3 \cdot 3 + 5$

Step 2.1.2.1.3.4

Move $3$ to the left of $h$ .

$h (t + h) = t^{2} + t h + 3 t + h t + h^{2} + 3 \cdot h + 3 t + 3 h + 3 \cdot 3 + 5$

Step 2.1.2.1.3.5

Multiply $3$ by $3$ .

$h (t + h) = t^{2} + t h + 3 t + h t + h^{2} + 3 h + 3 t + 3 h + 9 + 5$

Step 2.1.2.1.4

Add $t h$ and $h t$ .

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Step 2.1.2.1.4.1

Reorder $t$ and $h$ .

$h (t + h) = t^{2} + h t + h t + 3 t + h^{2} + 3 h + 3 t + 3 h + 9 + 5$

Step 2.1.2.1.4.2

Add $h t$ and $h t$ .

$h (t + h) = t^{2} + 2 h t + 3 t + h^{2} + 3 h + 3 t + 3 h + 9 + 5$

Step 2.1.2.1.5

Add $3 t$ and $3 t$ .

$h (t + h) = t^{2} + 2 h t + h^{2} + 3 h + 6 t + 3 h + 9 + 5$

Step 2.1.2.1.6

Add $3 h$ and $3 h$ .

$h (t + h) = t^{2} + 2 h t + h^{2} + 6 h + 6 t + 9 + 5$

Step 2.1.2.2

Add $9$ and $5$ .

$h (t + h) = t^{2} + 2 h t + h^{2} + 6 h + 6 t + 14$

Step 2.1.2.3

The final answer is $t^{2} + 2 h t + h^{2} + 6 h + 6 t + 14$ .

$t^{2} + 2 h t + h^{2} + 6 h + 6 t + 14$

Step 2.2

Reorder.

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Step 2.2.1

Move $t^{2}$ .

$2 h t + h^{2} + t^{2} + 6 h + 6 t + 14$

Step 2.2.2

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

$h^{2} + 2 h t + t^{2} + 6 h + 6 t + 14$

Step 2.3

Find the components of the definition.

$h (t + h) = h^{2} + 2 h t + t^{2} + 6 h + 6 t + 14$

$h (t) = t^{2} + 6 t + 14$

$h (t + h) = h^{2} + 2 h t + t^{2} + 6 h + 6 t + 14$

$h (t) = t^{2} + 6 t + 14$

Step 3

Plug in the components.

$\frac{h (t + h) - h t}{h} = \frac{h^{2} + 2 h t + t^{2} + 6 h + 6 t + 14 - (t^{2} + 6 t + 14)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{h^{2} + 2 h t + t^{2} + 6 h + 6 t + 14 - t^{2} - (6 t) - 1 \cdot 14}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $6$ by $- 1$ .

$\frac{h^{2} + 2 h t + t^{2} + 6 h + 6 t + 14 - t^{2} - 6 t - 1 \cdot 14}{h}$

Step 4.1.2.2

Multiply $- 1$ by $14$ .

$\frac{h^{2} + 2 h t + t^{2} + 6 h + 6 t + 14 - t^{2} - 6 t - 14}{h}$

Step 4.1.3

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

$\frac{h^{2} + 2 h t + 6 h + 6 t + 14 + 0 - 6 t - 14}{h}$

Step 4.1.4

Add $h^{2}$ and $0$ .

$\frac{h^{2} + 2 h t + 6 h + 6 t + 14 - 6 t - 14}{h}$

Step 4.1.5

Subtract $6 t$ from $6 t$ .

$\frac{h^{2} + 2 h t + 6 h + 0 + 14 - 14}{h}$

Step 4.1.6

Add $h^{2}$ and $0$ .

$\frac{h^{2} + 2 h t + 6 h + 14 - 14}{h}$

Step 4.1.7

Subtract $14$ from $14$ .

$\frac{h^{2} + 2 h t + 6 h + 0}{h}$

Step 4.1.8

Add $h^{2} + 2 h t + 6 h$ and $0$ .

$\frac{h^{2} + 2 h t + 6 h}{h}$

Step 4.1.9

Factor $h$ out of $h^{2} + 2 h t + 6 h$ .

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Step 4.1.9.1

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

$\frac{h \cdot h + 2 h t + 6 h}{h}$

Step 4.1.9.2

Factor $h$ out of $2 h t$ .

$\frac{h (h) + h (2 t) + 6 h}{h}$

Step 4.1.9.3

Factor $h$ out of $6 h$ .

$\frac{h (h) + h (2 t) + h \cdot 6}{h}$

Step 4.1.9.4

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

$\frac{h (h + 2 t) + h \cdot 6}{h}$

Step 4.1.9.5

Factor $h$ out of $h (h + 2 t) + h \cdot 6$ .

$\frac{h (h + 2 t + 6)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{h (h + 2 t + 6)}{h}$

Step 4.2.1.2

Divide $h + 2 t + 6$ by $1$ .

$h + 2 t + 6$

Step 4.2.2

Reorder $h$ and $2 t$ .

$2 t + h + 6$

Step 5