Algebra Examples

$f (x) = \sqrt{x} - 5 x$

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.

$f (x + h) = \sqrt{x + h} - 5 (x + h)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \sqrt{x + h} - 5 (x + h)$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = \sqrt{x + h} - 5 x - 5 h$

Step 2.1.2.3

The final answer is $\sqrt{x + h} - 5 x - 5 h$ .

$\sqrt{x + h} - 5 x - 5 h$

Step 2.2

Reorder.

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Step 2.2.1

Reorder $x$ and $h$ .

$\sqrt{h + x} - 5 x - 5 h$

Step 2.2.2

Move $\sqrt{h + x}$ .

$- 5 x - 5 h + \sqrt{h + x}$

Step 2.2.3

Reorder $- 5 x$ and $- 5 h$ .

$- 5 h - 5 x + \sqrt{h + x}$

Step 2.3

Find the components of the definition.

$f (x + h) = - 5 h - 5 x + \sqrt{h + x}$

$f (x) = \sqrt{x} - 5 x$

$f (x + h) = - 5 h - 5 x + \sqrt{h + x}$

$f (x) = \sqrt{x} - 5 x$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 5 h - 5 x + \sqrt{h + x} - (\sqrt{x} - 5 x)}{h}$

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.

$\frac{- 5 h - 5 x + \sqrt{h + x} - \sqrt{x} - (- 5 x)}{h}$

Step 4.1.2

Multiply $- 5$ by $- 1$ .

$\frac{- 5 h - 5 x + \sqrt{h + x} - \sqrt{x} + 5 x}{h}$

Step 4.1.3

Add $- 5 x$ and $5 x$ .

$\frac{- 5 h + 0 + \sqrt{h + x} - \sqrt{x}}{h}$

Step 4.1.4

Add $- 5 h$ and $0$ .

$\frac{- 5 h + \sqrt{h + x} - \sqrt{x}}{h}$

Step 4.2

Simplify with factoring out.

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Step 4.2.1

Factor $- 1$ out of $- 5 h$ .

$\frac{- (5 h) + \sqrt{h + x} - \sqrt{x}}{h}$

Step 4.2.2

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

$\frac{- (5 h) - 1 (- \sqrt{h + x}) - \sqrt{x}}{h}$

Step 4.2.3

Factor $- 1$ out of $- (5 h) - 1 (- \sqrt{h + x})$ .

$\frac{- (5 h - \sqrt{h + x}) - \sqrt{x}}{h}$

Step 4.2.4

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

$\frac{- (5 h - \sqrt{h + x}) - (\sqrt{x})}{h}$

Step 4.2.5

Factor $- 1$ out of $- (5 h - \sqrt{h + x}) - (\sqrt{x})$ .

$\frac{- (5 h - \sqrt{h + x} + \sqrt{x})}{h}$

Step 4.2.6

Simplify the expression.

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Step 4.2.6.1

Rewrite $- (5 h - \sqrt{h + x} + \sqrt{x})$ as $- 1 (5 h - \sqrt{h + x} + \sqrt{x})$ .

$\frac{- 1 (5 h - \sqrt{h + x} + \sqrt{x})}{h}$

Step 4.2.6.2

Move the negative in front of the fraction.

$- \frac{5 h - \sqrt{h + x} + \sqrt{x}}{h}$

Step 5