Precalculus Examples

$y = 5 x^{4} - 4 x^{3} + 25 x + 8000$

Step 1

Write $y = 5 x^{4} - 4 x^{3} + 25 x + 8000$ as a function.

$f (x) = 5 x^{4} - 4 x^{3} + 25 x + 8000$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = 5 {(x + h)}^{4} - 4 {(x + h)}^{3} + 25 (x + h) + 8000$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Use the Binomial Theorem.

$f (x + h) = 5 (x^{4} + 4 x^{3} h + 6 x^{2} h^{2} + 4 x h^{3} + h^{4}) - 4 {(x + h)}^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.2

Apply the distributive property.

$f (x + h) = 5 x^{4} + 5 (4 x^{3} h) + 5 (6 x^{2} h^{2}) + 5 (4 x h^{3}) + 5 h^{4} - 4 {(x + h)}^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.3

Simplify.

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Step 3.1.2.1.3.1

Multiply $4$ by $5$ .

$f (x + h) = 5 x^{4} + 20 (x^{3} h) + 5 (6 x^{2} h^{2}) + 5 (4 x h^{3}) + 5 h^{4} - 4 {(x + h)}^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.3.2

Multiply $6$ by $5$ .

$f (x + h) = 5 x^{4} + 20 (x^{3} h) + 30 (x^{2} h^{2}) + 5 (4 x h^{3}) + 5 h^{4} - 4 {(x + h)}^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.3.3

Multiply $4$ by $5$ .

$f (x + h) = 5 x^{4} + 20 (x^{3} h) + 30 (x^{2} h^{2}) + 20 (x h^{3}) + 5 h^{4} - 4 {(x + h)}^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.4

Remove parentheses.

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 {(x + h)}^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.5

Use the Binomial Theorem.

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 (x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3}) + 25 (x + h) + 8000$

Step 3.1.2.1.6

Apply the distributive property.

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 4 (3 x^{2} h) - 4 (3 x h^{2}) - 4 h^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.7

Simplify.

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Step 3.1.2.1.7.1

Multiply $3$ by $- 4$ .

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 (x^{2} h) - 4 (3 x h^{2}) - 4 h^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.7.2

Multiply $3$ by $- 4$ .

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 (x^{2} h) - 12 (x h^{2}) - 4 h^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.8

Remove parentheses.

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 x^{2} h - 12 x h^{2} - 4 h^{3} + 25 (x + h) + 8000$

Step 3.1.2.1.9

Apply the distributive property.

$f (x + h) = 5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 x^{2} h - 12 x h^{2} - 4 h^{3} + 25 x + 25 h + 8000$

Step 3.1.2.2

The final answer is $5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 x^{2} h - 12 x h^{2} - 4 h^{3} + 25 x + 25 h + 8000$ .

$5 x^{4} + 20 x^{3} h + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 x^{2} h - 12 x h^{2} - 4 h^{3} + 25 x + 25 h + 8000$

Step 3.2

Reorder.

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Step 3.2.1

Move $x^{3}$ .

$5 x^{4} + 20 h x^{3} + 30 x^{2} h^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 x^{2} h - 12 x h^{2} - 4 h^{3} + 25 x + 25 h + 8000$

Step 3.2.2

Move $x^{2}$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 x h^{3} + 5 h^{4} - 4 x^{3} - 12 x^{2} h - 12 x h^{2} - 4 h^{3} + 25 x + 25 h + 8000$

Step 3.2.3

Move $x$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} - 4 x^{3} - 12 x^{2} h - 12 x h^{2} - 4 h^{3} + 25 x + 25 h + 8000$

Step 3.2.4

Move $x^{2}$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} - 4 x^{3} - 12 h x^{2} - 12 x h^{2} - 4 h^{3} + 25 x + 25 h + 8000$

Step 3.2.5

Move $x$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} - 4 x^{3} - 12 h x^{2} - 12 h^{2} x - 4 h^{3} + 25 x + 25 h + 8000$

Step 3.2.6

Move $25 x$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} - 4 x^{3} - 12 h x^{2} - 12 h^{2} x - 4 h^{3} + 25 h + 25 x + 8000$

Step 3.2.7

Move $- 4 x^{3}$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} - 12 h x^{2} - 12 h^{2} x - 4 h^{3} - 4 x^{3} + 25 h + 25 x + 8000$

Step 3.2.8

Move $- 12 h x^{2}$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} - 12 h^{2} x - 4 h^{3} - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

Step 3.2.9

Move $- 12 h^{2} x$ .

$5 x^{4} + 20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

Step 3.2.10

Move $5 x^{4}$ .

$20 h x^{3} + 30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

Step 3.2.11

Move $20 h x^{3}$ .

$30 h^{2} x^{2} + 20 h^{3} x + 5 h^{4} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

Step 3.2.12

Move $30 h^{2} x^{2}$ .

$20 h^{3} x + 5 h^{4} + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

Step 3.2.13

Reorder $20 h^{3} x$ and $5 h^{4}$ .

$5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

Step 3.3

Find the components of the definition.

$f (x + h) = 5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

$f (x) = 5 x^{4} - 4 x^{3} + 25 x + 8000$

$f (x + h) = 5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000$

$f (x) = 5 x^{4} - 4 x^{3} + 25 x + 8000$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 - (5 x^{4} - 4 x^{3} + 25 x + 8000)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 - (5 x^{4}) - (- 4 x^{3}) - (25 x) - 1 \cdot 8000}{h}$

Step 5.1.2

Simplify.

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Step 5.1.2.1

Multiply $5$ by $- 1$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 - 5 x^{4} - (- 4 x^{3}) - (25 x) - 1 \cdot 8000}{h}$

Step 5.1.2.2

Multiply $- 4$ by $- 1$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 - 5 x^{4} + 4 x^{3} - (25 x) - 1 \cdot 8000}{h}$

Step 5.1.2.3

Multiply $25$ by $- 1$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 - 5 x^{4} + 4 x^{3} - 25 x - 1 \cdot 8000}{h}$

Step 5.1.2.4

Multiply $- 1$ by $8000$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} + 5 x^{4} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 - 5 x^{4} + 4 x^{3} - 25 x - 8000}{h}$

Step 5.1.3

Subtract $5 x^{4}$ from $5 x^{4}$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 + 0 + 4 x^{3} - 25 x - 8000}{h}$

Step 5.1.4

Add $5 h^{4}$ and $0$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} - 4 x^{3} + 25 h + 25 x + 8000 + 4 x^{3} - 25 x - 8000}{h}$

Step 5.1.5

Add $- 4 x^{3}$ and $4 x^{3}$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h + 25 x + 8000 + 0 - 25 x - 8000}{h}$

Step 5.1.6

Add $5 h^{4}$ and $0$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h + 25 x + 8000 - 25 x - 8000}{h}$

Step 5.1.7

Subtract $25 x$ from $25 x$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h + 0 + 8000 - 8000}{h}$

Step 5.1.8

Add $5 h^{4}$ and $0$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h + 8000 - 8000}{h}$

Step 5.1.9

Subtract $8000$ from $8000$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h + 0}{h}$

Step 5.1.10

Add $5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h$ and $0$ .

$\frac{5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h}{h}$

Step 5.1.11

Factor $h$ out of $5 h^{4} + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h$ .

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Step 5.1.11.1

Factor $h$ out of $5 h^{4}$ .

$\frac{h (5 h^{3}) + 20 h^{3} x + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h}{h}$

Step 5.1.11.2

Factor $h$ out of $20 h^{3} x$ .

$\frac{h (5 h^{3}) + h (20 h^{2} x) + 30 h^{2} x^{2} + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h}{h}$

Step 5.1.11.3

Factor $h$ out of $30 h^{2} x^{2}$ .

$\frac{h (5 h^{3}) + h (20 h^{2} x) + h (30 h x^{2}) + 20 h x^{3} - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h}{h}$

Step 5.1.11.4

Factor $h$ out of $20 h x^{3}$ .

$\frac{h (5 h^{3}) + h (20 h^{2} x) + h (30 h x^{2}) + h (20 x^{3}) - 4 h^{3} - 12 h^{2} x - 12 h x^{2} + 25 h}{h}$

Step 5.1.11.5

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

$\frac{h (5 h^{3}) + h (20 h^{2} x) + h (30 h x^{2}) + h (20 x^{3}) + h (- 4 h^{2}) - 12 h^{2} x - 12 h x^{2} + 25 h}{h}$

Step 5.1.11.6

Factor $h$ out of $- 12 h^{2} x$ .

$\frac{h (5 h^{3}) + h (20 h^{2} x) + h (30 h x^{2}) + h (20 x^{3}) + h (- 4 h^{2}) + h (- 12 h x) - 12 h x^{2} + 25 h}{h}$

Step 5.1.11.7

Factor $h$ out of $- 12 h x^{2}$ .

$\frac{h (5 h^{3}) + h (20 h^{2} x) + h (30 h x^{2}) + h (20 x^{3}) + h (- 4 h^{2}) + h (- 12 h x) + h (- 12 x^{2}) + 25 h}{h}$

Step 5.1.11.8

Factor $h$ out of $25 h$ .

$\frac{h (5 h^{3}) + h (20 h^{2} x) + h (30 h x^{2}) + h (20 x^{3}) + h (- 4 h^{2}) + h (- 12 h x) + h (- 12 x^{2}) + h \cdot 25}{h}$

Step 5.1.11.9

Factor $h$ out of $h (5 h^{3}) + h (20 h^{2} x)$ .

$\frac{h (5 h^{3} + 20 h^{2} x) + h (30 h x^{2}) + h (20 x^{3}) + h (- 4 h^{2}) + h (- 12 h x) + h (- 12 x^{2}) + h \cdot 25}{h}$

Step 5.1.11.10

Factor $h$ out of $h (5 h^{3} + 20 h^{2} x) + h (30 h x^{2})$ .

$\frac{h (5 h^{3} + 20 h^{2} x + 30 h x^{2}) + h (20 x^{3}) + h (- 4 h^{2}) + h (- 12 h x) + h (- 12 x^{2}) + h \cdot 25}{h}$

Step 5.1.11.11

Factor $h$ out of $h (5 h^{3} + 20 h^{2} x + 30 h x^{2}) + h (20 x^{3})$ .

$\frac{h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3}) + h (- 4 h^{2}) + h (- 12 h x) + h (- 12 x^{2}) + h \cdot 25}{h}$

Step 5.1.11.12

Factor $h$ out of $h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3}) + h (- 4 h^{2})$ .

$\frac{h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2}) + h (- 12 h x) + h (- 12 x^{2}) + h \cdot 25}{h}$

Step 5.1.11.13

Factor $h$ out of $h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2}) + h (- 12 h x)$ .

$\frac{h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x) + h (- 12 x^{2}) + h \cdot 25}{h}$

Step 5.1.11.14

Factor $h$ out of $h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x) + h (- 12 x^{2})$ .

$\frac{h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2}) + h \cdot 25}{h}$

Step 5.1.11.15

Factor $h$ out of $h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2}) + h \cdot 25$ .

$\frac{h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2} + 25)}{h}$

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Cancel the common factor of $h$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{h (5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2} + 25)}{h}$

Step 5.2.1.2

Divide $5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2} + 25$ by $1$ .

$5 h^{3} + 20 h^{2} x + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2} + 25$

Step 5.2.2

Simplify the expression.

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Step 5.2.2.1

Move $h^{2}$ .

$5 h^{3} + 20 x h^{2} + 30 h x^{2} + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2} + 25$

Step 5.2.2.2

Move $h$ .

$5 h^{3} + 20 x h^{2} + 30 x^{2} h + 20 x^{3} - 4 h^{2} - 12 h x - 12 x^{2} + 25$

Step 5.2.2.3

Move $h$ .

$5 h^{3} + 20 x h^{2} + 30 x^{2} h + 20 x^{3} - 4 h^{2} - 12 x h - 12 x^{2} + 25$

Step 5.2.2.4

Move $- 4 h^{2}$ .

$5 h^{3} + 20 x h^{2} + 30 x^{2} h + 20 x^{3} - 12 x h - 12 x^{2} - 4 h^{2} + 25$

Step 5.2.2.5

Move $- 12 x h$ .

$5 h^{3} + 20 x h^{2} + 30 x^{2} h + 20 x^{3} - 12 x^{2} - 12 x h - 4 h^{2} + 25$

Step 5.2.2.6

Move $5 h^{3}$ .

$20 x h^{2} + 30 x^{2} h + 20 x^{3} + 5 h^{3} - 12 x^{2} - 12 x h - 4 h^{2} + 25$

Step 5.2.2.7

Move $20 x h^{2}$ .

$30 x^{2} h + 20 x^{3} + 20 x h^{2} + 5 h^{3} - 12 x^{2} - 12 x h - 4 h^{2} + 25$

Step 5.2.2.8

Reorder $30 x^{2} h$ and $20 x^{3}$ .

$20 x^{3} + 30 x^{2} h + 20 x h^{2} + 5 h^{3} - 12 x^{2} - 12 x h - 4 h^{2} + 25$

Step 6