Calculus Examples

$f (x) = \frac{1}{5} \cdot (145 - x)$

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.

$f (x + h) = \frac{1}{5} \cdot (145 - (x + h))$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = \frac{1}{5} \cdot (145 - x - h)$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = \frac{1}{5} \cdot 145 + \frac{1}{5} \cdot (- x) + \frac{1}{5} \cdot (- h)$

Step 2.1.2.3

Simplify.

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Step 2.1.2.3.1

Cancel the common factor of $5$ .

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Step 2.1.2.3.1.1

Factor $5$ out of $145$ .

$f (x + h) = \frac{1}{5} \cdot (5 (29)) + \frac{1}{5} \cdot (- x) + \frac{1}{5} \cdot (- h)$

Step 2.1.2.3.1.2

Cancel the common factor.

$f (x + h) = \frac{1}{5} \cdot (5 \cdot 29) + \frac{1}{5} \cdot (- x) + \frac{1}{5} \cdot (- h)$

Step 2.1.2.3.1.3

Rewrite the expression.

$f (x + h) = 29 + \frac{1}{5} \cdot (- x) + \frac{1}{5} \cdot (- h)$

Step 2.1.2.3.2

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

$f (x + h) = 29 - \frac{x}{5} + \frac{1}{5} \cdot (- h)$

Step 2.1.2.3.3

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

$f (x + h) = 29 - \frac{x}{5} - \frac{h}{5}$

Step 2.1.2.4

The final answer is $29 - \frac{x}{5} - \frac{h}{5}$ .

$29 - \frac{x}{5} - \frac{h}{5}$

Step 2.2

Reorder.

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Step 2.2.1

Move $29$ .

$- \frac{x}{5} - \frac{h}{5} + 29$

Step 2.2.2

Reorder $- \frac{x}{5}$ and $- \frac{h}{5}$ .

$- \frac{h}{5} - \frac{x}{5} + 29$

Step 2.3

Find the components of the definition.

$f (x + h) = - \frac{h}{5} - \frac{x}{5} + 29$

$f (x) = 29 - \frac{x}{5}$

$f (x + h) = - \frac{h}{5} - \frac{x}{5} + 29$

$f (x) = 29 - \frac{x}{5}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{h}{5} - \frac{x}{5} + 29 - (29 - \frac{x}{5})}{h}$

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.

$\frac{- \frac{h}{5} - \frac{x}{5} + 29 - 1 \cdot 29 - - \frac{x}{5}}{h}$

Step 4.1.2

Multiply $- 1$ by $29$ .

$\frac{- \frac{h}{5} - \frac{x}{5} + 29 - 29 - - \frac{x}{5}}{h}$

Step 4.1.3

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

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Step 4.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{h}{5} - \frac{x}{5} + 29 - 29 + 1 \frac{x}{5}}{h}$

Step 4.1.3.2

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

$\frac{- \frac{h}{5} - \frac{x}{5} + 29 - 29 + \frac{x}{5}}{h}$

Step 4.1.4

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

$\frac{- \frac{h}{5} + 0 + 29 - 29}{h}$

Step 4.1.5

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

$\frac{- \frac{h}{5} + 29 - 29}{h}$

Step 4.1.6

Subtract $29$ from $29$ .

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

Step 4.1.7

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

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

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

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

Step 4.3.2

Factor $h$ out of $- h$ .

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

$\frac{- 1}{5}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{1}{5}$

Step 5