Calculus Examples

$f (x) = \frac{2}{x - 3}$

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.

$f (x + h) = \frac{2}{(x + h) - 3}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{2}{x + h - 3}$

Step 2.1.2.2

The final answer is $\frac{2}{x + h - 3}$ .

$\frac{2}{x + h - 3}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{2}{x + h - 3}$

$f (x) = \frac{2}{x - 3}$

$f (x + h) = \frac{2}{x + h - 3}$

$f (x) = \frac{2}{x - 3}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{2}{x + h - 3} - (\frac{2}{x - 3})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{2}{x + h - 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{\frac{2}{x + h - 3} \cdot \frac{x - 3}{x - 3} - \frac{2}{x - 3}}{h}$

Step 4.1.2

To write $- \frac{2}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + h - 3}{x + h - 3}$ .

$\frac{\frac{2}{x + h - 3} \cdot \frac{x - 3}{x - 3} - \frac{2}{x - 3} \cdot \frac{x + h - 3}{x + h - 3}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(x + h - 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{2}{x + h - 3}$ by $\frac{x - 3}{x - 3}$ .

$\frac{\frac{2 (x - 3)}{(x + h - 3) (x - 3)} - \frac{2}{x - 3} \cdot \frac{x + h - 3}{x + h - 3}}{h}$

Step 4.1.3.2

Multiply $\frac{2}{x - 3}$ by $\frac{x + h - 3}{x + h - 3}$ .

$\frac{\frac{2 (x - 3)}{(x + h - 3) (x - 3)} - \frac{2 (x + h - 3)}{(x - 3) (x + h - 3)}}{h}$

Step 4.1.3.3

Reorder the factors of $(x - 3) (x + h - 3)$ .

$\frac{\frac{2 (x - 3)}{(x + h - 3) (x - 3)} - \frac{2 (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{2 (x - 3) - 2 (x + h - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5

Rewrite $\frac{2 (x - 3) - 2 (x + h - 3)}{(x + h - 3) (x - 3)}$ in a factored form.

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Step 4.1.5.1

Factor $2$ out of $2 (x - 3) - 2 (x + h - 3)$ .

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Step 4.1.5.1.1

Factor $2$ out of $- 2 (x + h - 3)$ .

$\frac{\frac{2 (x - 3) + 2 (- (x + h - 3))}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.1.2

Factor $2$ out of $2 (x - 3) + 2 (- (x + h - 3))$ .

$\frac{\frac{2 (x - 3 - (x + h - 3))}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.2

Apply the distributive property.

$\frac{\frac{2 (x - 3 - x - h - - 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.3

Multiply $- 1$ by $- 3$ .

$\frac{\frac{2 (x - 3 - x - h + 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.4

Subtract $x$ from $x$ .

$\frac{\frac{2 (0 - 3 - h + 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.5

Subtract $3$ from $0$ .

$\frac{\frac{2 (- 3 - h + 3)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.6

Add $- 3$ and $3$ .

$\frac{\frac{2 (- h + 0)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.7

Add $- h$ and $0$ .

$\frac{\frac{2 \cdot - 1 h}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.8

Combine exponents.

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Step 4.1.5.8.1

Factor out negative.

$\frac{\frac{- (2 h)}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.5.8.2

Multiply $2$ by $- 1$ .

$\frac{\frac{- 2 h}{(x + h - 3) (x - 3)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{2 h}{(x + h - 3) (x - 3)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{2 h}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{2 h}{(x + h - 3) (x - 3)}$ into the numerator.

$\frac{- 2 h}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 2 h$ .

$\frac{h \cdot - 2}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 2}{(x + h - 3) (x - 3)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 2}{(x + h - 3) (x - 3)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{2}{(x + h - 3) (x - 3)}$

Step 5