Precalculus Examples

f (x) = 7 x^{2} + 5 x - 4

$f (x) = 7 x^{2} + 5 x - 4$

Step 1

Consider the difference quotient formula.

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

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

Step 2

Find the components of the definition.

Tap for more steps...

Step 2.1

Evaluate the function at

x = x + h

$x = x + h$ .

Tap for more steps...

Step 2.1.1

Replace the variable

x

$x$ with

x + h

$x + h$ in the expression.

f (x + h) = 7 {(x + h)}^{2} + 5 (x + h) - 4

$f (x + h) = 7 {(x + h)}^{2} + 5 (x + h) - 4$

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}

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

(x + h) (x + h)

$(x + h) (x + h)$ .

f (x + h) = 7 ((x + h) (x + h)) + 5 (x + h) - 4

$f (x + h) = 7 ((x + h) (x + h)) + 5 (x + h) - 4$

Step 2.1.2.1.2

Expand

(x + h) (x + h)

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

Tap for more steps...

Step 2.1.2.1.2.1

Apply the distributive property.

f (x + h) = 7 (x (x + h) + h (x + h)) + 5 (x + h) - 4

$f (x + h) = 7 (x (x + h) + h (x + h)) + 5 (x + h) - 4$

Step 2.1.2.1.2.2

Apply the distributive property.

f (x + h) = 7 (x \cdot x + x h + h (x + h)) + 5 (x + h) - 4

$f (x + h) = 7 (x \cdot x + x h + h (x + h)) + 5 (x + h) - 4$

Step 2.1.2.1.2.3

Apply the distributive property.

f (x + h) = 7 (x \cdot x + x h + h x + h \cdot h) + 5 (x + h) - 4

$f (x + h) = 7 (x \cdot x + x h + h x + h \cdot h) + 5 (x + h) - 4$

f (x + h) = 7 (x \cdot x + x h + h x + h \cdot h) + 5 (x + h) - 4

$f (x + h) = 7 (x \cdot x + x h + h x + h \cdot h) + 5 (x + h) - 4$

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

$x$ by

x

$x$ .

f (x + h) = 7 (x^{2} + x h + h x + h \cdot h) + 5 (x + h) - 4

$f (x + h) = 7 (x^{2} + x h + h x + h \cdot h) + 5 (x + h) - 4$

Step 2.1.2.1.3.1.2

Multiply

h

$h$ by

h

$h$ .

f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 5 (x + h) - 4

$f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 5 (x + h) - 4$

f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 5 (x + h) - 4

$f (x + h) = 7 (x^{2} + x h + h x + h^{2}) + 5 (x + h) - 4$

Step 2.1.2.1.3.2

Add

x h

$x h$ and

h x

$h x$ .

Tap for more steps...

Step 2.1.2.1.3.2.1

Reorder

x

$x$ and

h

$h$ .

f (x + h) = 7 (x^{2} + h x + h x + h^{2}) + 5 (x + h) - 4

$f (x + h) = 7 (x^{2} + h x + h x + h^{2}) + 5 (x + h) - 4$

Step 2.1.2.1.3.2.2

Add

h x

$h x$ and

h x

$h x$ .

f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 5 (x + h) - 4

$f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 5 (x + h) - 4$

f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 5 (x + h) - 4

$f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 5 (x + h) - 4$

f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 5 (x + h) - 4

$f (x + h) = 7 (x^{2} + 2 h x + h^{2}) + 5 (x + h) - 4$

Step 2.1.2.1.4

Apply the distributive property.

f (x + h) = 7 x^{2} + 7 (2 h x) + 7 h^{2} + 5 (x + h) - 4

$f (x + h) = 7 x^{2} + 7 (2 h x) + 7 h^{2} + 5 (x + h) - 4$

Step 2.1.2.1.5

Multiply

2

$2$ by

7

$7$ .

f (x + h) = 7 x^{2} + 14 (h x) + 7 h^{2} + 5 (x + h) - 4

$f (x + h) = 7 x^{2} + 14 (h x) + 7 h^{2} + 5 (x + h) - 4$

Step 2.1.2.1.6

Apply the distributive property.

f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4

$f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4$

f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4

$f (x + h) = 7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4$

Step 2.1.2.2

The final answer is

7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4

$7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4$ .

7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4

$7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4$

7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4

$7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4$

7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4

$7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4$

Step 2.2

Reorder.

Tap for more steps...

Step 2.2.1

Move

5 x

$5 x$ .

7 x^{2} + 14 h x + 7 h^{2} + 5 h + 5 x - 4

$7 x^{2} + 14 h x + 7 h^{2} + 5 h + 5 x - 4$

Step 2.2.2

Move

7 x^{2}

$7 x^{2}$ .

14 h x + 7 h^{2} + 7 x^{2} + 5 h + 5 x - 4

$14 h x + 7 h^{2} + 7 x^{2} + 5 h + 5 x - 4$

Step 2.2.3

Reorder

14 h x

$14 h x$ and

7 h^{2}

$7 h^{2}$ .

7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4

$7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4$

7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4

$7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4$

Step 2.3

Find the components of the definition.

f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4

$f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4$

f (x) = 7 x^{2} + 5 x - 4

$f (x) = 7 x^{2} + 5 x - 4$

f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4

$f (x + h) = 7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4$

f (x) = 7 x^{2} + 5 x - 4

$f (x) = 7 x^{2} + 5 x - 4$

Step 3

Plug in the components.

\frac{f (x + h) - f (x)}{h} = \frac{7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4 - (7 x^{2} + 5 x - 4)}{h}

$\frac{f (x + h) - f (x)}{h} = \frac{7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4 - (7 x^{2} + 5 x - 4)}{h}$

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{7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4 - (7 x^{2}) - (5 x) - - 4}{h}

$\frac{7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4 - (7 x^{2}) - (5 x) - - 4}{h}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Multiply

$7$ by

$- 1$ .

$\frac{7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4 - 7 x^{2} - (5 x) - - 4}{h}$

Step 4.1.2.2

Multiply

$5$ by

$- 1$ .

$\frac{7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4 - 7 x^{2} - 5 x - - 4}{h}$

Step 4.1.2.3

Multiply

$- 1$ by

$- 4$ .

$\frac{7 h^{2} + 14 h x + 7 x^{2} + 5 h + 5 x - 4 - 7 x^{2} - 5 x + 4}{h}$

Step 4.1.3

Subtract

$7 x^{2}$ from

$7 x^{2}$ .

$\frac{7 h^{2} + 14 h x + 5 h + 5 x - 4 + 0 - 5 x + 4}{h}$

Step 4.1.4

Add

$7 h^{2}$ and

$0$ .

$\frac{7 h^{2} + 14 h x + 5 h + 5 x - 4 - 5 x + 4}{h}$

Step 4.1.5

Subtract

$5 x$ from

$5 x$ .

$\frac{7 h^{2} + 14 h x + 5 h + 0 - 4 + 4}{h}$

Step 4.1.6

Add

$7 h^{2}$ and

$0$ .

$\frac{7 h^{2} + 14 h x + 5 h - 4 + 4}{h}$

Step 4.1.7

Add

$- 4$ and

$4$ .

$\frac{7 h^{2} + 14 h x + 5 h + 0}{h}$

Step 4.1.8

Add

$7 h^{2} + 14 h x + 5 h$ and

$0$ .

$\frac{7 h^{2} + 14 h x + 5 h}{h}$

Step 4.1.9

Factor

$h$ out of

$7 h^{2} + 14 h x + 5 h$ .

Tap for more steps...

Step 4.1.9.1

Factor

$h$ out of

$7 h^{2}$ .

$\frac{h (7 h) + 14 h x + 5 h}{h}$

Step 4.1.9.2

Factor

$h$ out of

$14 h x$ .

$\frac{h (7 h) + h (14 x) + 5 h}{h}$

Step 4.1.9.3

Factor

$h$ out of

$5 h$ .

$\frac{h (7 h) + h (14 x) + h \cdot 5}{h}$

Step 4.1.9.4

Factor

$h$ out of

$h (7 h) + h (14 x)$ .

$\frac{h (7 h + 14 x) + h \cdot 5}{h}$

Step 4.1.9.5

Factor

$h$ out of

$h (7 h + 14 x) + h \cdot 5$ .

$\frac{h (7 h + 14 x + 5)}{h}$

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.

$\frac{h (7 h + 14 x + 5)}{h}$

Step 4.2.1.2

Divide

$7 h + 14 x + 5$ by

$1$ .

$7 h + 14 x + 5$

Step 4.2.2

Reorder

$7 h$ and

$14 x$ .

$14 x + 7 h + 5$

Step 5

Enter YOUR Problem