Calculus Examples

$f (x) = \frac{6}{x + 10}$

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.

$f (x + h) = \frac{6}{(x + h) + 10}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{6}{x + h + 10}$

Step 2.1.2.2

The final answer is $\frac{6}{x + h + 10}$ .

$\frac{6}{x + h + 10}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{6}{x + h + 10}$

$f (x) = \frac{6}{x + 10}$

$f (x + h) = \frac{6}{x + h + 10}$

$f (x) = \frac{6}{x + 10}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{6}{x + h + 10} - (\frac{6}{x + 10})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{6}{x + h + 10}$ as a fraction with a common denominator, multiply by $\frac{x + 10}{x + 10}$ .

$\frac{\frac{6}{x + h + 10} \cdot \frac{x + 10}{x + 10} - \frac{6}{x + 10}}{h}$

Step 4.1.2

To write $- \frac{6}{x + 10}$ as a fraction with a common denominator, multiply by $\frac{x + h + 10}{x + h + 10}$ .

$\frac{\frac{6}{x + h + 10} \cdot \frac{x + 10}{x + 10} - \frac{6}{x + 10} \cdot \frac{x + h + 10}{x + h + 10}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h + 10) (x + 10)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{6}{x + h + 10}$ by $\frac{x + 10}{x + 10}$ .

$\frac{\frac{6 (x + 10)}{(x + h + 10) (x + 10)} - \frac{6}{x + 10} \cdot \frac{x + h + 10}{x + h + 10}}{h}$

Step 4.1.3.2

Multiply $\frac{6}{x + 10}$ by $\frac{x + h + 10}{x + h + 10}$ .

$\frac{\frac{6 (x + 10)}{(x + h + 10) (x + 10)} - \frac{6 (x + h + 10)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x + h + 10) (x + 10)$ .

$\frac{\frac{6 (x + 10)}{(x + 10) (x + h + 10)} - \frac{6 (x + h + 10)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{6 (x + 10) - 6 (x + h + 10)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5

Rewrite $\frac{6 (x + 10) - 6 (x + h + 10)}{(x + 10) (x + h + 10)}$ in a factored form.

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Step 4.1.5.1

Factor $6$ out of $6 (x + 10) - 6 (x + h + 10)$ .

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Step 4.1.5.1.1

Factor $6$ out of $- 6 (x + h + 10)$ .

$\frac{\frac{6 (x + 10) + 6 (- (x + h + 10))}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.1.2

Factor $6$ out of $6 (x + 10) + 6 (- (x + h + 10))$ .

$\frac{\frac{6 (x + 10 - (x + h + 10))}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.2

Apply the distributive property.

$\frac{\frac{6 (x + 10 - x - h - 1 \cdot 10)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.3

Multiply $- 1$ by $10$ .

$\frac{\frac{6 (x + 10 - x - h - 10)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.4

Subtract $x$ from $x$ .

$\frac{\frac{6 (0 + 10 - h - 10)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.5

Add $0$ and $10$ .

$\frac{\frac{6 (10 - h - 10)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.6

Subtract $10$ from $10$ .

$\frac{\frac{6 (- h + 0)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.7

Add $- h$ and $0$ .

$\frac{\frac{6 \cdot - 1 h}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.8

Combine exponents.

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Step 4.1.5.8.1

Factor out negative.

$\frac{\frac{- (6 h)}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.5.8.2

Multiply $6$ by $- 1$ .

$\frac{\frac{- 6 h}{(x + 10) (x + h + 10)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{6 h}{(x + 10) (x + h + 10)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{6 h}{(x + 10) (x + h + 10)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{6 h}{(x + 10) (x + h + 10)}$ into the numerator.

$\frac{- 6 h}{(x + 10) (x + h + 10)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 6 h$ .

$\frac{h \cdot - 6}{(x + 10) (x + h + 10)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 6}{(x + 10) (x + h + 10)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 6}{(x + 10) (x + h + 10)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{6}{(x + 10) (x + h + 10)}$

Step 5