Precalculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Rewrite ${(x + h - 1)}^{2}$ as $(x + h - 1) (x + h - 1)$ .

$f (x + h) = (x + h - 1) (x + h - 1)$

Step 2.1.2.2

Expand $(x + h - 1) (x + h - 1)$ by multiplying each term in the first expression by each term in the second expression.

$f (x + h) = x \cdot x + x h + x \cdot - 1 + h x + h \cdot h + h \cdot - 1 - 1 x - 1 h - 1 \cdot - 1$

Step 2.1.2.3

Simplify each term.

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Step 2.1.2.3.1

Multiply $x$ by $x$ .

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

Step 2.1.2.3.2

Move $- 1$ to the left of $x$ .

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

Step 2.1.2.3.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.2.3.4

Multiply $h$ by $h$ .

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

Step 2.1.2.3.5

Move $- 1$ to the left of $h$ .

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

Step 2.1.2.3.6

Rewrite $- 1 h$ as $- h$ .

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

Step 2.1.2.3.7

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.2.3.8

Rewrite $- 1 h$ as $- h$ .

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

Step 2.1.2.3.9

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.4

Add $x h$ and $h x$ .

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Step 2.1.2.4.1

Reorder $x$ and $h$ .

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

Step 2.1.2.4.2

Add $h x$ and $h x$ .

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

Step 2.1.2.5

Subtract $x$ from $- x$ .

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

Step 2.1.2.6

Subtract $h$ from $- h$ .

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

Step 2.1.2.7

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

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

Step 2.2

Reorder.

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Step 2.2.1

Move $x^{2}$ .

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

Step 2.2.2

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

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

Step 2.3

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

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.

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- 2$ by $- 1$ .

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

Step 4.1.2.2

Multiply $- 1$ by $1$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

Add $- 2 x$ and $2 x$ .

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

Step 4.1.6

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

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

Step 4.1.7

Subtract $1$ from $1$ .

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

Step 4.1.8

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

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

Step 4.1.9

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

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Step 4.1.9.1

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

$\frac{h \cdot h + 2 h x - 2 h}{h}$

Step 4.1.9.2

Factor $h$ out of $2 h x$ .

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

Step 4.1.9.3

Factor $h$ out of $- 2 h$ .

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

Step 4.1.9.4

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

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

Step 4.1.9.5

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

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

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.

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

Step 4.2.1.2

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

$h + 2 x - 2$

Step 4.2.2

Reorder $h$ and $2 x$ .

$2 x + h - 2$

Step 5