Precalculus Examples

$f (x) = x^{3} - 3 x^{2} + 2 x$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$f (x + h) = {(x + h)}^{3} - 3 {(x + h)}^{2} + 2 (x + h)$

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.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 {(x + h)}^{2} + 2 (x + h)$

Step 2.1.2.1.2

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

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 ((x + h) (x + h)) + 2 (x + h)$

Step 2.1.2.1.3

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

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Step 2.1.2.1.3.1

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 (x (x + h) + h (x + h)) + 2 (x + h)$

Step 2.1.2.1.3.2

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 (x \cdot x + x h + h (x + h)) + 2 (x + h)$

Step 2.1.2.1.3.3

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 (x \cdot x + x h + h x + h \cdot h) + 2 (x + h)$

Step 2.1.2.1.4

Simplify and combine like terms.

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Step 2.1.2.1.4.1

Simplify each term.

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Step 2.1.2.1.4.1.1

Multiply $x$ by $x$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 (x^{2} + x h + h x + h \cdot h) + 2 (x + h)$

Step 2.1.2.1.4.1.2

Multiply $h$ by $h$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 (x^{2} + x h + h x + h^{2}) + 2 (x + h)$

Step 2.1.2.1.4.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.4.2.1

Reorder $x$ and $h$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 (x^{2} + h x + h x + h^{2}) + 2 (x + h)$

Step 2.1.2.1.4.2.2

Add $h x$ and $h x$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 (x^{2} + 2 h x + h^{2}) + 2 (x + h)$

Step 2.1.2.1.5

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 x^{2} - 3 (2 h x) - 3 h^{2} + 2 (x + h)$

Step 2.1.2.1.6

Multiply $2$ by $- 3$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 x^{2} - 6 (h x) - 3 h^{2} + 2 (x + h)$

Step 2.1.2.1.7

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 x^{2} - 6 h x - 3 h^{2} + 2 x + 2 h$

Step 2.1.2.2

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

$x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 3 x^{2} - 6 h x - 3 h^{2} + 2 x + 2 h$

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

$x^{3} + 3 h x^{2} + 3 x h^{2} + h^{3} - 3 x^{2} - 6 h x - 3 h^{2} + 2 x + 2 h$

Step 2.2.2

Move $x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 3 x^{2} - 6 h x - 3 h^{2} + 2 x + 2 h$

Step 2.2.3

Move $2 x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 3 x^{2} - 6 h x - 3 h^{2} + 2 h + 2 x$

Step 2.2.4

Move $- 3 x^{2}$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 6 h x - 3 h^{2} - 3 x^{2} + 2 h + 2 x$

Step 2.2.5

Move $- 6 h x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 3 h^{2} - 6 h x - 3 x^{2} + 2 h + 2 x$

Step 2.2.6

Move $x^{3}$ .

$3 h x^{2} + 3 h^{2} x + h^{3} + x^{3} - 3 h^{2} - 6 h x - 3 x^{2} + 2 h + 2 x$

Step 2.2.7

Move $3 h x^{2}$ .

$3 h^{2} x + h^{3} + 3 h x^{2} + x^{3} - 3 h^{2} - 6 h x - 3 x^{2} + 2 h + 2 x$

Step 2.2.8

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

$h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 3 h^{2} - 6 h x - 3 x^{2} + 2 h + 2 x$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 3 h^{2} - 6 h x - 3 x^{2} + 2 h + 2 x$

$f (x) = x^{3} - 3 x^{2} + 2 x$

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 3 h^{2} - 6 h x - 3 x^{2} + 2 h + 2 x$

$f (x) = x^{3} - 3 x^{2} + 2 x$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 3 h^{2} - 6 h x - 3 x^{2} + 2 h + 2 x - (x^{3} - 3 x^{2} + 2 x)}{h}$

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.

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- 3$ by $- 1$ .

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

Step 4.1.2.2

Multiply $2$ by $- 1$ .

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

Step 4.1.3

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

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

Step 4.1.4

Add $h^{3}$ and $0$ .

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

Step 4.1.5

Add $- 3 x^{2}$ and $3 x^{2}$ .

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

Step 4.1.6

Add $h^{3}$ and $0$ .

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

Step 4.1.7

Subtract $2 x$ from $2 x$ .

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

Step 4.1.8

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

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

Step 4.1.9

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

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Step 4.1.9.1

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

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

Step 4.1.9.2

Factor $h$ out of $3 h^{2} x$ .

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

Step 4.1.9.3

Factor $h$ out of $3 h x^{2}$ .

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

Step 4.1.9.4

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

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

Step 4.1.9.5

Factor $h$ out of $- 6 h x$ .

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

Step 4.1.9.6

Factor $h$ out of $2 h$ .

$\frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h (- 3 h) + h (- 6 x) + h \cdot 2}{h}$

Step 4.1.9.7

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

$\frac{h (h^{2} + 3 h x) + h (3 x^{2}) + h (- 3 h) + h (- 6 x) + h \cdot 2}{h}$

Step 4.1.9.8

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

$\frac{h (h^{2} + 3 h x + 3 x^{2}) + h (- 3 h) + h (- 6 x) + h \cdot 2}{h}$

Step 4.1.9.9

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

$\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h) + h (- 6 x) + h \cdot 2}{h}$

Step 4.1.9.10

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

$\frac{h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x) + h \cdot 2}{h}$

Step 4.1.9.11

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x) + h \cdot 2$ .

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

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.

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

Step 4.2.1.2

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

$h^{2} + 3 h x + 3 x^{2} - 3 h - 6 x + 2$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Move $h$ .

$h^{2} + 3 x h + 3 x^{2} - 3 h - 6 x + 2$

Step 4.2.2.2

Move $- 3 h$ .

$h^{2} + 3 x h + 3 x^{2} - 6 x - 3 h + 2$

Step 4.2.2.3

Move $h^{2}$ .

$3 x h + 3 x^{2} + h^{2} - 6 x - 3 h + 2$

Step 4.2.2.4

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

$3 x^{2} + 3 x h + h^{2} - 6 x - 3 h + 2$

Step 5