Precalculus Examples

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

Step 1

Evaluate the numerator in $\frac{f (x + h) - f (x)}{h}$ .

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Step 1.1

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

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

Step 1.2

Replace the function designators in $f (x + h) - f (x)$ with the actual functions.

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

Step 1.3

Simplify $(2 {(x + h)}^{2} - (x + h) - 3) - (2 x^{2} - x - 3)$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Apply the distributive property.

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

Step 1.3.1.2.2

Apply the distributive property.

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

Step 1.3.1.2.3

Apply the distributive property.

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

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.3.1.3.1.2

Multiply $h$ by $h$ .

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

Step 1.3.1.3.2

Add $x h$ and $h x$ .

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Step 1.3.1.3.2.1

Reorder $x$ and $h$ .

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

Step 1.3.1.3.2.2

Add $h x$ and $h x$ .

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

Step 1.3.1.4

Apply the distributive property.

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

Step 1.3.1.5

Multiply $2$ by $2$ .

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

Step 1.3.1.6

Apply the distributive property.

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

Step 1.3.1.7

Apply the distributive property.

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

Step 1.3.1.8

Simplify.

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Step 1.3.1.8.1

Multiply $2$ by $- 1$ .

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

Step 1.3.1.8.2

Multiply $- - x$ .

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Step 1.3.1.8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.1.8.2.2

Multiply $x$ by $1$ .

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

Step 1.3.1.8.3

Multiply $- 1$ by $- 3$ .

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

Step 1.3.2

Combine the opposite terms in $2 x^{2} + 4 h x + 2 h^{2} - x - h - 3 - 2 x^{2} + x + 3$ .

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Step 1.3.2.1

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

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Add $- x$ and $x$ .

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

Step 1.3.2.4

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

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

Step 1.3.2.5

Add $- 3$ and $3$ .

$4 h x + 2 h^{2} - h + 0$

Step 1.3.2.6

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Factor $h$ out of $- h$ .

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

Step 3.1.4

Factor $h$ out of $h (4 x) + h (2 h)$ .

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

Step 3.1.5

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

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

Step 3.2

Cancel the common factor of $h$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Divide $4 x + 2 h - 1$ by $1$ .

$4 x + 2 h - 1$

Step 3.3

Reorder $4 x$ and $2 h$ .

$2 h + 4 x - 1$