Precalculus Examples

$f (x) = 12 x^{2} - 3$

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.

$f (x + h) = 12 {(x + h)}^{2} - 3$

Step 2.1.2

Simplify the result.

Tap for more steps...

Step 2.1.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.1

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

$f (x + h) = 12 ((x + h) (x + h)) - 3$

Step 2.1.2.1.2

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

Tap for more steps...

Step 2.1.2.1.2.1

Apply the distributive property.

$f (x + h) = 12 (x (x + h) + h (x + h)) - 3$

Step 2.1.2.1.2.2

Apply the distributive property.

$f (x + h) = 12 (x \cdot x + x h + h (x + h)) - 3$

Step 2.1.2.1.2.3

Apply the distributive property.

$f (x + h) = 12 (x \cdot x + x h + h x + h \cdot h) - 3$

Step 2.1.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

$f (x + h) = 12 (x^{2} + x h + h x + h \cdot h) - 3$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$f (x + h) = 12 (x^{2} + x h + h x + h^{2}) - 3$

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

Tap for more steps...

Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

$f (x + h) = 12 (x^{2} + h x + h x + h^{2}) - 3$

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

$f (x + h) = 12 (x^{2} + 2 h x + h^{2}) - 3$

Step 2.1.2.1.4

Apply the distributive property.

$f (x + h) = 12 x^{2} + 12 (2 h x) + 12 h^{2} - 3$

Step 2.1.2.1.5

Multiply $2$ by $12$ .

$f (x + h) = 12 x^{2} + 24 h x + 12 h^{2} - 3$

Step 2.1.2.2

The final answer is $12 x^{2} + 24 h x + 12 h^{2} - 3$ .

$12 x^{2} + 24 h x + 12 h^{2} - 3$

Step 2.2

Reorder.

Tap for more steps...

Step 2.2.1

Move $12 x^{2}$ .

$24 h x + 12 h^{2} + 12 x^{2} - 3$

Step 2.2.2

Reorder $24 h x$ and $12 h^{2}$ .

$12 h^{2} + 24 h x + 12 x^{2} - 3$

Step 2.3

Find the components of the definition.

$f (x + h) = 12 h^{2} + 24 h x + 12 x^{2} - 3$

$f (x) = 12 x^{2} - 3$

$f (x + h) = 12 h^{2} + 24 h x + 12 x^{2} - 3$

$f (x) = 12 x^{2} - 3$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - (12 x^{2} - 3)}{h}$

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.

$\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - (12 x^{2}) - - 3}{h}$

Step 4.1.2

Multiply $12$ by $- 1$ .

$\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - 12 x^{2} - - 3}{h}$

Step 4.1.3

Multiply $- 1$ by $- 3$ .

$\frac{12 h^{2} + 24 h x + 12 x^{2} - 3 - 12 x^{2} + 3}{h}$

Step 4.1.4

Subtract $12 x^{2}$ from $12 x^{2}$ .

$\frac{12 h^{2} + 24 h x + 0 - 3 + 3}{h}$

Step 4.1.5

Add $12 h^{2}$ and $0$ .

$\frac{12 h^{2} + 24 h x - 3 + 3}{h}$

Step 4.1.6

Add $- 3$ and $3$ .

$\frac{12 h^{2} + 24 h x + 0}{h}$

Step 4.1.7

Add $12 h^{2} + 24 h x$ and $0$ .

$\frac{12 h^{2} + 24 h x}{h}$

Step 4.1.8

Factor $12 h$ out of $12 h^{2} + 24 h x$ .

Tap for more steps...

Step 4.1.8.1

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

$\frac{12 h \cdot h + 24 h x}{h}$

Step 4.1.8.2

Factor $12 h$ out of $24 h x$ .

$\frac{12 h \cdot h + 12 h (2 x)}{h}$

Step 4.1.8.3

Factor $12 h$ out of $12 h \cdot h + 12 h (2 x)$ .

$\frac{12 h (h + 2 x)}{h}$

Step 4.2

Simplify terms.

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.

$\frac{12 h (h + 2 x)}{h}$

Step 4.2.1.2

Divide $12 (h + 2 x)$ by $1$ .

$12 (h + 2 x)$

Step 4.2.2

Apply the distributive property.

$12 h + 12 (2 x)$

Step 4.2.3

Simplify the expression.

Tap for more steps...

Step 4.2.3.1

Multiply $2$ by $12$ .

$12 h + 24 x$

Step 4.2.3.2

Reorder $12 h$ and $24 x$ .

$24 x + 12 h$

Step 5

Enter YOUR Problem