Precalculus Examples

$f (x) = \sqrt{x} - x + 1$

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.

$f (x + h) = \sqrt{x + h} - (x + h) + 1$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \sqrt{x + h} - (x + h) + 1$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = \sqrt{x + h} - x - h + 1$

Step 2.1.2.3

The final answer is $\sqrt{x + h} - x - h + 1$ .

$\sqrt{x + h} - x - h + 1$

Step 2.2

Reorder.

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Step 2.2.1

Reorder $x$ and $h$ .

$\sqrt{h + x} - x - h + 1$

Step 2.2.2

Move $\sqrt{h + x}$ .

$- x - h + \sqrt{h + x} + 1$

Step 2.2.3

Reorder $- x$ and $- h$ .

$- h - x + \sqrt{h + x} + 1$

Step 2.3

Find the components of the definition.

$f (x + h) = - h - x + \sqrt{h + x} + 1$

$f (x) = \sqrt{x} - x + 1$

$f (x + h) = - h - x + \sqrt{h + x} + 1$

$f (x) = \sqrt{x} - x + 1$

Step 3

Plug in the components.

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

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.

$\frac{- h - x + \sqrt{h + x} + 1 - \sqrt{x} - - x - 1 \cdot 1}{h}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- - x$ .

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Step 4.1.2.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- h - x + \sqrt{h + x} + 1 - \sqrt{x} + 1 x - 1 \cdot 1}{h}$

Step 4.1.2.1.2

Multiply $x$ by $1$ .

$\frac{- h - x + \sqrt{h + x} + 1 - \sqrt{x} + x - 1 \cdot 1}{h}$

Step 4.1.2.2

Multiply $- 1$ by $1$ .

$\frac{- h - x + \sqrt{h + x} + 1 - \sqrt{x} + x - 1}{h}$

Step 4.1.3

Add $- x$ and $x$ .

$\frac{- h + 0 + \sqrt{h + x} + 1 - \sqrt{x} - 1}{h}$

Step 4.1.4

Add $- h$ and $0$ .

$\frac{- h + \sqrt{h + x} + 1 - \sqrt{x} - 1}{h}$

Step 4.1.5

Subtract $1$ from $1$ .

$\frac{- h + \sqrt{h + x} + 0 - \sqrt{x}}{h}$

Step 4.1.6

Add $- h$ and $0$ .

$\frac{- h + \sqrt{h + x} - \sqrt{x}}{h}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Factor $- 1$ out of $\sqrt{h + x}$ .

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

Step 4.2.2

Factor $- 1$ out of $- h - 1 (- \sqrt{h + x})$ .

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

Step 4.2.3

Factor $- 1$ out of $- \sqrt{x}$ .

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

Step 4.2.4

Factor $- 1$ out of $- (h - \sqrt{h + x}) - (\sqrt{x})$ .

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

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

Rewrite $- (h - \sqrt{h + x} + \sqrt{x})$ as $- 1 (h - \sqrt{h + x} + \sqrt{x})$ .

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

Step 4.2.5.2

Move the negative in front of the fraction.

$- \frac{h - \sqrt{h + x} + \sqrt{x}}{h}$

Step 5