Precalculus Examples

$g (t) = t^{4} - t^{3} + t^{2}$

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.

$g (t + h) = {(t + h)}^{4} - {(t + h)}^{3} + {(t + h)}^{2}$

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

Use the Binomial Theorem.

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - {(t + h)}^{3} + {(t + h)}^{2}$

Step 2.1.2.1.2

Use the Binomial Theorem.

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - (t^{3} + 3 t^{2} h + 3 t h^{2} + h^{3}) + {(t + h)}^{2}$

Step 2.1.2.1.3

Apply the distributive property.

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - (3 t^{2} h) - (3 t h^{2}) - h^{3} + {(t + h)}^{2}$

Step 2.1.2.1.4

Simplify.

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Step 2.1.2.1.4.1

Multiply $3$ by $- 1$ .

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 (t^{2} h) - (3 t h^{2}) - h^{3} + {(t + h)}^{2}$

Step 2.1.2.1.4.2

Multiply $3$ by $- 1$ .

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 (t^{2} h) - 3 (t h^{2}) - h^{3} + {(t + h)}^{2}$

Step 2.1.2.1.5

Remove parentheses.

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + {(t + h)}^{2}$

Step 2.1.2.1.6

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

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + (t + h) (t + h)$

Step 2.1.2.1.7

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

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Step 2.1.2.1.7.1

Apply the distributive property.

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t (t + h) + h (t + h)$

Step 2.1.2.1.7.2

Apply the distributive property.

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t \cdot t + t h + h (t + h)$

Step 2.1.2.1.7.3

Apply the distributive property.

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t \cdot t + t h + h t + h \cdot h$

Step 2.1.2.1.8

Simplify and combine like terms.

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Step 2.1.2.1.8.1

Simplify each term.

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Step 2.1.2.1.8.1.1

Multiply $t$ by $t$ .

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + t h + h t + h \cdot h$

Step 2.1.2.1.8.1.2

Multiply $h$ by $h$ .

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + t h + h t + h^{2}$

Step 2.1.2.1.8.2

Add $t h$ and $h t$ .

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Step 2.1.2.1.8.2.1

Reorder $t$ and $h$ .

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + h t + h t + h^{2}$

Step 2.1.2.1.8.2.2

Add $h t$ and $h t$ .

$g (t + h) = t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + 2 h t + h^{2}$

Step 2.1.2.2

The final answer is $t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + 2 h t + h^{2}$ .

$t^{4} + 4 t^{3} h + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + 2 h t + h^{2}$

Step 2.2

Reorder.

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Step 2.2.1

Move $t^{3}$ .

$t^{4} + 4 h t^{3} + 6 t^{2} h^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + 2 h t + h^{2}$

Step 2.2.2

Move $t^{2}$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 t h^{3} + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + 2 h t + h^{2}$

Step 2.2.3

Move $t$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - t^{3} - 3 t^{2} h - 3 t h^{2} - h^{3} + t^{2} + 2 h t + h^{2}$

Step 2.2.4

Move $t^{2}$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - t^{3} - 3 h t^{2} - 3 t h^{2} - h^{3} + t^{2} + 2 h t + h^{2}$

Step 2.2.5

Move $t$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - t^{3} - 3 h t^{2} - 3 h^{2} t - h^{3} + t^{2} + 2 h t + h^{2}$

Step 2.2.6

Move $t^{2}$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - t^{3} - 3 h t^{2} - 3 h^{2} t - h^{3} + 2 h t + h^{2} + t^{2}$

Step 2.2.7

Move $2 h t$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - t^{3} - 3 h t^{2} - 3 h^{2} t - h^{3} + h^{2} + 2 h t + t^{2}$

Step 2.2.8

Move $- t^{3}$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - 3 h t^{2} - 3 h^{2} t - h^{3} - t^{3} + h^{2} + 2 h t + t^{2}$

Step 2.2.9

Move $- 3 h t^{2}$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - 3 h^{2} t - h^{3} - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

Step 2.2.10

Move $- 3 h^{2} t$ .

$t^{4} + 4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

Step 2.2.11

Move $t^{4}$ .

$4 h t^{3} + 6 h^{2} t^{2} + 4 h^{3} t + h^{4} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

Step 2.2.12

Move $4 h t^{3}$ .

$6 h^{2} t^{2} + 4 h^{3} t + h^{4} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

Step 2.2.13

Move $6 h^{2} t^{2}$ .

$4 h^{3} t + h^{4} + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

Step 2.2.14

Reorder $4 h^{3} t$ and $h^{4}$ .

$h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

Step 2.3

Find the components of the definition.

$g (t + h) = h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

$g (t) = t^{4} - t^{3} + t^{2}$

$g (t + h) = h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2}$

$g (t) = t^{4} - t^{3} + t^{2}$

Step 3

Plug in the components.

$\frac{g (t + h) - g t}{h} = \frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2} - (t^{4} - t^{3} + t^{2})}{h}$

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.

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2} - t^{4} - - t^{3} - t^{2}}{h}$

Step 4.1.2

Multiply $- - t^{3}$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2} - t^{4} + 1 t^{3} - t^{2}}{h}$

Step 4.1.2.2

Multiply $t^{3}$ by $1$ .

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} + t^{4} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2} - t^{4} + t^{3} - t^{2}}{h}$

Step 4.1.3

Subtract $t^{4}$ from $t^{4}$ .

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2} + 0 + t^{3} - t^{2}}{h}$

Step 4.1.4

Add $h^{4}$ and $0$ .

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} - t^{3} + h^{2} + 2 h t + t^{2} + t^{3} - t^{2}}{h}$

Step 4.1.5

Add $- t^{3}$ and $t^{3}$ .

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t + t^{2} + 0 - t^{2}}{h}$

Step 4.1.6

Add $h^{4}$ and $0$ .

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t + t^{2} - t^{2}}{h}$

Step 4.1.7

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

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t + 0}{h}$

Step 4.1.8

Add $h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t$ and $0$ .

$\frac{h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t}{h}$

Step 4.1.9

Factor $h$ out of $h^{4} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t$ .

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Step 4.1.9.1

Factor $h$ out of $h^{4}$ .

$\frac{h \cdot h^{3} + 4 h^{3} t + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t}{h}$

Step 4.1.9.2

Factor $h$ out of $4 h^{3} t$ .

$\frac{h (h^{3}) + h (4 h^{2} t) + 6 h^{2} t^{2} + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t}{h}$

Step 4.1.9.3

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

$\frac{h (h^{3}) + h (4 h^{2} t) + h (6 h t^{2}) + 4 h t^{3} - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t}{h}$

Step 4.1.9.4

Factor $h$ out of $4 h t^{3}$ .

$\frac{h (h^{3}) + h (4 h^{2} t) + h (6 h t^{2}) + h (4 t^{3}) - h^{3} - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t}{h}$

Step 4.1.9.5

Factor $h$ out of $- h^{3}$ .

$\frac{h (h^{3}) + h (4 h^{2} t) + h (6 h t^{2}) + h (4 t^{3}) + h (- h^{2}) - 3 h^{2} t - 3 h t^{2} + h^{2} + 2 h t}{h}$

Step 4.1.9.6

Factor $h$ out of $- 3 h^{2} t$ .

$\frac{h (h^{3}) + h (4 h^{2} t) + h (6 h t^{2}) + h (4 t^{3}) + h (- h^{2}) + h (- 3 h t) - 3 h t^{2} + h^{2} + 2 h t}{h}$

Step 4.1.9.7

Factor $h$ out of $- 3 h t^{2}$ .

$\frac{h (h^{3}) + h (4 h^{2} t) + h (6 h t^{2}) + h (4 t^{3}) + h (- h^{2}) + h (- 3 h t) + h (- 3 t^{2}) + h^{2} + 2 h t}{h}$

Step 4.1.9.8

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

$\frac{h (h^{3}) + h (4 h^{2} t) + h (6 h t^{2}) + h (4 t^{3}) + h (- h^{2}) + h (- 3 h t) + h (- 3 t^{2}) + h (h) + 2 h t}{h}$

Step 4.1.9.9

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

$\frac{h (h^{3}) + h (4 h^{2} t) + h (6 h t^{2}) + h (4 t^{3}) + h (- h^{2}) + h (- 3 h t) + h (- 3 t^{2}) + h (h) + h (2 t)}{h}$

Step 4.1.9.10

Factor $h$ out of $h (h^{3}) + h (4 h^{2} t)$ .

$\frac{h (h^{3} + 4 h^{2} t) + h (6 h t^{2}) + h (4 t^{3}) + h (- h^{2}) + h (- 3 h t) + h (- 3 t^{2}) + h (h) + h (2 t)}{h}$

Step 4.1.9.11

Factor $h$ out of $h (h^{3} + 4 h^{2} t) + h (6 h t^{2})$ .

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2}) + h (4 t^{3}) + h (- h^{2}) + h (- 3 h t) + h (- 3 t^{2}) + h (h) + h (2 t)}{h}$

Step 4.1.9.12

Factor $h$ out of $h (h^{3} + 4 h^{2} t + 6 h t^{2}) + h (4 t^{3})$ .

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3}) + h (- h^{2}) + h (- 3 h t) + h (- 3 t^{2}) + h (h) + h (2 t)}{h}$

Step 4.1.9.13

Factor $h$ out of $h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3}) + h (- h^{2})$ .

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2}) + h (- 3 h t) + h (- 3 t^{2}) + h (h) + h (2 t)}{h}$

Step 4.1.9.14

Factor $h$ out of $h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2}) + h (- 3 h t)$ .

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t) + h (- 3 t^{2}) + h (h) + h (2 t)}{h}$

Step 4.1.9.15

Factor $h$ out of $h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t) + h (- 3 t^{2})$ .

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2}) + h (h) + h (2 t)}{h}$

Step 4.1.9.16

Factor $h$ out of $h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2}) + h (h)$ .

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h) + h (2 t)}{h}$

Step 4.1.9.17

Factor $h$ out of $h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h) + h (2 t)$ .

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h + 2 t)}{h}$

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.

$\frac{h (h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h + 2 t)}{h}$

Step 4.2.1.2

Divide $h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h + 2 t$ by $1$ .

$h^{3} + 4 h^{2} t + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h + 2 t$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h^{2}$ .

$h^{3} + 4 t h^{2} + 6 h t^{2} + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h + 2 t$

Step 4.2.2.2

Move $h$ .

$h^{3} + 4 t h^{2} + 6 t^{2} h + 4 t^{3} - h^{2} - 3 h t - 3 t^{2} + h + 2 t$

Step 4.2.2.3

Move $h$ .

$h^{3} + 4 t h^{2} + 6 t^{2} h + 4 t^{3} - h^{2} - 3 t h - 3 t^{2} + h + 2 t$

Step 4.2.2.4

Move $h$ .

$h^{3} + 4 t h^{2} + 6 t^{2} h + 4 t^{3} - h^{2} - 3 t h - 3 t^{2} + 2 t + h$

Step 4.2.2.5

Move $- h^{2}$ .

$h^{3} + 4 t h^{2} + 6 t^{2} h + 4 t^{3} - 3 t h - 3 t^{2} - h^{2} + 2 t + h$

Step 4.2.2.6

Move $- 3 t h$ .

$h^{3} + 4 t h^{2} + 6 t^{2} h + 4 t^{3} - 3 t^{2} - 3 t h - h^{2} + 2 t + h$

Step 4.2.2.7

Move $h^{3}$ .

$4 t h^{2} + 6 t^{2} h + 4 t^{3} + h^{3} - 3 t^{2} - 3 t h - h^{2} + 2 t + h$

Step 4.2.2.8

Move $4 t h^{2}$ .

$6 t^{2} h + 4 t^{3} + 4 t h^{2} + h^{3} - 3 t^{2} - 3 t h - h^{2} + 2 t + h$

Step 4.2.2.9

Reorder $6 t^{2} h$ and $4 t^{3}$ .

$4 t^{3} + 6 t^{2} h + 4 t h^{2} + h^{3} - 3 t^{2} - 3 t h - h^{2} + 2 t + h$

Step 5