Precalculus Examples

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

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.

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

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

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

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

Step 2.1.2.1.2

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

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Step 2.1.2.1.2.1

Apply the distributive property.

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

Step 2.1.2.1.2.2

Apply the distributive property.

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

Step 2.1.2.1.2.3

Apply the distributive property.

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

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

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

Step 2.1.2.1.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.1.3.2.1

Reorder $x$ and $h$ .

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

Step 2.1.2.1.3.2.2

Add $h x$ and $h x$ .

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

Step 2.1.2.1.4

Apply the distributive property.

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

Step 2.1.2.1.5

Multiply $2$ by $7$ .

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

Step 2.1.2.1.6

Apply the distributive property.

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

Step 2.1.2.2

The final answer is $7 x^{2} + 14 h x + 7 h^{2} + 5 x + 5 h - 4$ .

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

Step 2.2

Reorder.

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Step 2.2.1

Move $5 x$ .

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

Step 2.2.2

Move $7 x^{2}$ .

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

Step 2.2.3

Reorder $14 h x$ and $7 h^{2}$ .

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Simplify.

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

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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.

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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.

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

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

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