Precalculus Examples

$y = - \frac{1}{5} x$

Step 1

Write $y = - \frac{1}{5} x$ as a function.

$f (x) = - \frac{1}{5} x$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = x + h$ .

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Step 3.1.1

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

$f (x + h) = - \frac{1}{5} \cdot (x + h)$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

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

Step 3.1.2.2

Combine $x$ and $\frac{1}{5}$ .

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

Step 3.1.2.3

Combine $h$ and $\frac{1}{5}$ .

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

Step 3.1.2.4

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

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

Step 3.2

Find the components of the definition.

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

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

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

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

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- - \frac{x}{5}$ .

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Step 5.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{x}{5} - \frac{h}{5} + 1 \frac{x}{5}}{h}$

Step 5.1.1.2

Multiply $\frac{x}{5}$ by $1$ .

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

Step 5.1.2

Add $- \frac{x}{5}$ and $\frac{x}{5}$ .

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

Step 5.1.3

Add $- \frac{h}{5}$ and $0$ .

$\frac{- \frac{h}{5}}{h}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{h}{5} \cdot \frac{1}{h}$

Step 5.3

Cancel the common factor of $h$ .

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Step 5.3.1

Move the leading negative in $- \frac{h}{5}$ into the numerator.

$\frac{- h}{5} \cdot \frac{1}{h}$

Step 5.3.2

Factor $h$ out of $- h$ .

$\frac{h \cdot - 1}{5} \cdot \frac{1}{h}$

Step 5.3.3

Cancel the common factor.

$\frac{h \cdot - 1}{5} \cdot \frac{1}{h}$

Step 5.3.4

Rewrite the expression.

$\frac{- 1}{5}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{1}{5}$

Step 6