Precalculus Examples

$f (x) = x^{3} - 2 x^{2} + x + 1$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

$f (x + h) = {(x + h)}^{3} - 2 {(x + h)}^{2} + x + h + 1$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Remove parentheses.

$f (x + h) = {(x + h)}^{3} - 2 {(x + h)}^{2} + x + h + 1$

Step 2.1.2.2

Simplify each term.

Tap for more steps...

Step 2.1.2.2.1

Use the Binomial Theorem.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 {(x + h)}^{2} + x + h + 1$

Step 2.1.2.2.2

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

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 ((x + h) (x + h)) + x + h + 1$

Step 2.1.2.2.3

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

Tap for more steps...

Step 2.1.2.2.3.1

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 (x (x + h) + h (x + h)) + x + h + 1$

Step 2.1.2.2.3.2

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 (x \cdot x + x h + h (x + h)) + x + h + 1$

Step 2.1.2.2.3.3

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 (x \cdot x + x h + h x + h \cdot h) + x + h + 1$

Step 2.1.2.2.4

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.2.4.1

Simplify each term.

Tap for more steps...

Step 2.1.2.2.4.1.1

Multiply $x$ by $x$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 (x^{2} + x h + h x + h \cdot h) + x + h + 1$

Step 2.1.2.2.4.1.2

Multiply $h$ by $h$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 (x^{2} + x h + h x + h^{2}) + x + h + 1$

Step 2.1.2.2.4.2

Add $x h$ and $h x$ .

Tap for more steps...

Step 2.1.2.2.4.2.1

Reorder $x$ and $h$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 (x^{2} + h x + h x + h^{2}) + x + h + 1$

Step 2.1.2.2.4.2.2

Add $h x$ and $h x$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 (x^{2} + 2 h x + h^{2}) + x + h + 1$

Step 2.1.2.2.5

Apply the distributive property.

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 x^{2} - 2 (2 h x) - 2 h^{2} + x + h + 1$

Step 2.1.2.2.6

Multiply $2$ by $- 2$ .

$f (x + h) = x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 x^{2} - 4 h x - 2 h^{2} + x + h + 1$

Step 2.1.2.3

The final answer is $x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 x^{2} - 4 h x - 2 h^{2} + x + h + 1$ .

$x^{3} + 3 x^{2} h + 3 x h^{2} + h^{3} - 2 x^{2} - 4 h x - 2 h^{2} + x + h + 1$

Step 2.2

Reorder.

Tap for more steps...

Step 2.2.1

Move $x^{2}$ .

$x^{3} + 3 h x^{2} + 3 x h^{2} + h^{3} - 2 x^{2} - 4 h x - 2 h^{2} + x + h + 1$

Step 2.2.2

Move $x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 2 x^{2} - 4 h x - 2 h^{2} + x + h + 1$

Step 2.2.3

Move $x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 2 x^{2} - 4 h x - 2 h^{2} + h + x + 1$

Step 2.2.4

Move $- 2 x^{2}$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 4 h x - 2 h^{2} - 2 x^{2} + h + x + 1$

Step 2.2.5

Move $- 4 h x$ .

$x^{3} + 3 h x^{2} + 3 h^{2} x + h^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1$

Step 2.2.6

Move $x^{3}$ .

$3 h x^{2} + 3 h^{2} x + h^{3} + x^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1$

Step 2.2.7

Move $3 h x^{2}$ .

$3 h^{2} x + h^{3} + 3 h x^{2} + x^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1$

Step 2.2.8

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

$h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1$

Step 2.3

Find the components of the definition.

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1$

$f (x) = x^{3} - 2 x^{2} + x + 1$

$f (x + h) = h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1$

$f (x) = x^{3} - 2 x^{2} + x + 1$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1 - (x^{3} - 2 x^{2} + x + 1)}{h}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Multiply $- 2$ by $- 1$ .

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} + x^{3} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1 - x^{3} + 2 x^{2} - x - 1 \cdot 1}{h}$

Step 4.1.2.2

Multiply $- 1$ by $1$ .

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

Step 4.1.3

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

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 2 h^{2} - 4 h x - 2 x^{2} + h + x + 1 + 0 + 2 x^{2} - x - 1}{h}$

Step 4.1.4

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

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

Step 4.1.5

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

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 2 h^{2} - 4 h x + h + x + 1 + 0 - x - 1}{h}$

Step 4.1.6

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

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

Step 4.1.7

Subtract $x$ from $x$ .

$\frac{h^{3} + 3 h^{2} x + 3 h x^{2} - 2 h^{2} - 4 h x + h + 0 + 1 - 1}{h}$

Step 4.1.8

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

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

Step 4.1.9

Subtract $1$ from $1$ .

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

Step 4.1.10

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

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

Step 4.1.11

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

Tap for more steps...

Step 4.1.11.1

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

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

Step 4.1.11.2

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

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

Step 4.1.11.3

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

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

Step 4.1.11.4

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

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

Step 4.1.11.5

Factor $h$ out of $- 4 h x$ .

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

Step 4.1.11.6

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

$\frac{h (h^{2}) + h (3 h x) + h (3 x^{2}) + h (- 2 h) + h (- 4 x) + h^{1}}{h}$

Step 4.1.11.7

Factor $h$ out of $h^{1}$ .

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

Step 4.1.11.8

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

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

Step 4.1.11.9

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

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

Step 4.1.11.10

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

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

Step 4.1.11.11

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

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

Step 4.1.11.12

Factor $h$ out of $h (h^{2} + 3 h x + 3 x^{2} - 2 h - 4 x) + h \cdot 1$ .

$\frac{h (h^{2} + 3 h x + 3 x^{2} - 2 h - 4 x + 1)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $h$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{h (h^{2} + 3 h x + 3 x^{2} - 2 h - 4 x + 1)}{h}$

Step 4.2.1.2

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

$h^{2} + 3 h x + 3 x^{2} - 2 h - 4 x + 1$

Step 4.2.2

Simplify the expression.

Tap for more steps...

Step 4.2.2.1

Move $h$ .

$h^{2} + 3 x h + 3 x^{2} - 2 h - 4 x + 1$

Step 4.2.2.2

Move $- 2 h$ .

$h^{2} + 3 x h + 3 x^{2} - 4 x - 2 h + 1$

Step 4.2.2.3

Move $h^{2}$ .

$3 x h + 3 x^{2} + h^{2} - 4 x - 2 h + 1$

Step 4.2.2.4

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

$3 x^{2} + 3 x h + h^{2} - 4 x - 2 h + 1$

Step 5