Calculus Examples

$f (x) = \frac{1 - x}{3 + x}$

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.

$f (x + h) = \frac{1 - (x + h)}{3 + x + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Remove parentheses.

$f (x + h) = \frac{1 - (x + h)}{3 + x + h}$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = \frac{1 - x - h}{3 + x + h}$

Step 2.1.2.3

The final answer is $\frac{1 - x - h}{3 + x + h}$ .

$\frac{1 - x - h}{3 + x + h}$

Step 2.2

Find the components of the definition.

$f (x + h) = \frac{1 - x - h}{3 + x + h}$

$f (x) = \frac{1 - x}{3 + x}$

$f (x + h) = \frac{1 - x - h}{3 + x + h}$

$f (x) = \frac{1 - x}{3 + x}$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{1 - x - h}{3 + x + h} - (\frac{1 - x}{3 + x})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{1 - x - h}{3 + x + h}$ as a fraction with a common denominator, multiply by $\frac{3 + x}{3 + x}$ .

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

Step 4.1.2

To write $- \frac{1 - x}{3 + x}$ as a fraction with a common denominator, multiply by $\frac{3 + x + h}{3 + x + h}$ .

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

Step 4.1.3

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

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Step 4.1.3.1

Multiply $\frac{1 - x - h}{3 + x + h}$ by $\frac{3 + x}{3 + x}$ .

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

Step 4.1.3.2

Multiply $\frac{1 - x}{3 + x}$ by $\frac{3 + x + h}{3 + x + h}$ .

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

Step 4.1.3.3

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

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Rewrite $\frac{(1 - x - h) (3 + x) - (1 - x) (3 + x + h)}{(3 + x) (3 + x + h)}$ in a factored form.

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Step 4.1.5.1

Expand $(1 - x - h) (3 + x)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.5.2

Simplify each term.

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Step 4.1.5.2.1

Multiply $3$ by $1$ .

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

Step 4.1.5.2.2

Multiply $x$ by $1$ .

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

Step 4.1.5.2.3

Multiply $3$ by $- 1$ .

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

Step 4.1.5.2.4

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.2.4.1

Move $x$ .

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

Step 4.1.5.2.4.2

Multiply $x$ by $x$ .

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

Step 4.1.5.2.5

Multiply $3$ by $- 1$ .

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

Step 4.1.5.3

Subtract $3 x$ from $x$ .

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

Step 4.1.5.4

Apply the distributive property.

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

Step 4.1.5.5

Multiply $- 1$ by $1$ .

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

Step 4.1.5.6

Multiply $- - x$ .

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Step 4.1.5.6.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.5.6.2

Multiply $x$ by $1$ .

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

Step 4.1.5.7

Expand $(- 1 + x) (3 + x + h)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.5.8

Simplify each term.

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Step 4.1.5.8.1

Multiply $- 1$ by $3$ .

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

Step 4.1.5.8.2

Rewrite $- 1 x$ as $- x$ .

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

Step 4.1.5.8.3

Rewrite $- 1 h$ as $- h$ .

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

Step 4.1.5.8.4

Move $3$ to the left of $x$ .

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

Step 4.1.5.8.5

Multiply $x$ by $x$ .

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

Step 4.1.5.9

Add $- x$ and $3 x$ .

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

Step 4.1.5.10

Subtract $3$ from $3$ .

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

Step 4.1.5.11

Add $- 2 x$ and $0$ .

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

Step 4.1.5.12

Add $- x^{2}$ and $x^{2}$ .

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

Step 4.1.5.13

Subtract $h$ from $- 3 h$ .

$\frac{\frac{- h x - 2 x + 2 x - 4 h + 0 + x h}{(3 + x) (3 + x + h)}}{h}$

Step 4.1.5.14

Add $- h x$ and $0$ .

$\frac{\frac{- h x - 2 x + 2 x - 4 h + x h}{(3 + x) (3 + x + h)}}{h}$

Step 4.1.5.15

Add $- h x$ and $x h$ .

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Step 4.1.5.15.1

Move $h$ .

$\frac{\frac{- 2 x + 2 x - 4 h - 1 x h + x h}{(3 + x) (3 + x + h)}}{h}$

Step 4.1.5.15.2

Add $- 1 x h$ and $x h$ .

$\frac{\frac{- 2 x + 2 x - 4 h + 0}{(3 + x) (3 + x + h)}}{h}$

Step 4.1.5.16

Add $- 2 x + 2 x - 4 h$ and $0$ .

$\frac{\frac{- 2 x + 2 x - 4 h}{(3 + x) (3 + x + h)}}{h}$

Step 4.1.5.17

Add $- 2 x$ and $2 x$ .

$\frac{\frac{0 - 4 h}{(3 + x) (3 + x + h)}}{h}$

Step 4.1.5.18

Subtract $4 h$ from $0$ .

$\frac{\frac{- 4 h}{(3 + x) (3 + x + h)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{4 h}{(3 + x) (3 + x + h)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{4 h}{(3 + x) (3 + x + h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{4 h}{(3 + x) (3 + x + h)}$ into the numerator.

$\frac{- 4 h}{(3 + x) (3 + x + h)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 4 h$ .

$\frac{h \cdot - 4}{(3 + x) (3 + x + h)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 4}{(3 + x) (3 + x + h)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 4}{(3 + x) (3 + x + h)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{4}{(3 + x) (3 + x + h)}$

Step 5