Calculus Examples

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

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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

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

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Write each expression with a common denominator of $(6 - x - h) (6 - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

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

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

Step 4.1.3.2

Multiply $\frac{x}{6 - x}$ by $\frac{6 - x - h}{6 - x - h}$ .

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

Step 4.1.3.3

Reorder the factors of $(6 - x - h) (6 - x)$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

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

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Step 4.1.5.1

Expand $(x + h) (6 - x)$ using the FOIL Method.

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Step 4.1.5.1.1

Apply the distributive property.

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

Step 4.1.5.1.2

Apply the distributive property.

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

Step 4.1.5.1.3

Apply the distributive property.

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

Step 4.1.5.2

Simplify each term.

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Step 4.1.5.2.1

Move $6$ to the left of $x$ .

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

Step 4.1.5.2.2

Rewrite using the commutative property of multiplication.

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

Step 4.1.5.2.3

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

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Step 4.1.5.2.3.1

Move $x$ .

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

Step 4.1.5.2.3.2

Multiply $x$ by $x$ .

$\frac{\frac{6 x - x^{2} + h \cdot 6 + h (- x) - x (6 - x - h)}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.2.4

Move $6$ to the left of $h$ .

$\frac{\frac{6 x - x^{2} + 6 \cdot h + h (- x) - x (6 - x - h)}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.2.5

Rewrite using the commutative property of multiplication.

$\frac{\frac{6 x - x^{2} + 6 h - h x - x (6 - x - h)}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.3

Apply the distributive property.

$\frac{\frac{6 x - x^{2} + 6 h - h x - x \cdot 6 - x (- x) - x (- h)}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.4

Simplify.

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Step 4.1.5.4.1

Multiply $6$ by $- 1$ .

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x - x (- x) - x (- h)}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.4.2

Rewrite using the commutative property of multiplication.

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x - 1 \cdot - 1 x \cdot x - x (- h)}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.4.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x - 1 \cdot - 1 x \cdot x - 1 \cdot - 1 x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.5

Simplify each term.

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Step 4.1.5.5.1

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

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Step 4.1.5.5.1.1

Move $x$ .

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x - 1 \cdot - 1 (x \cdot x) - 1 \cdot - 1 x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.5.1.2

Multiply $x$ by $x$ .

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x - 1 \cdot - 1 x^{2} - 1 \cdot - 1 x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.5.2

Multiply $- 1$ by $- 1$ .

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x + 1 x^{2} - 1 \cdot - 1 x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.5.3

Multiply $x^{2}$ by $1$ .

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x + x^{2} - 1 \cdot - 1 x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.5.4

Multiply $- 1$ by $- 1$ .

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x + x^{2} + 1 x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.5.5

Multiply $x$ by $1$ .

$\frac{\frac{6 x - x^{2} + 6 h - h x - 6 x + x^{2} + x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.6

Subtract $6 x$ from $6 x$ .

$\frac{\frac{- x^{2} + 6 h - h x + 0 + x^{2} + x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.7

Add $- x^{2}$ and $0$ .

$\frac{\frac{- x^{2} + 6 h - h x + x^{2} + x h}{(6 - x) (6 - x - h)}}{h}$

Step 4.1.5.8

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

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

Step 4.1.5.9

Add $6 h$ and $0$ .

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

Step 4.1.5.10

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

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Step 4.1.5.10.1

Move $h$ .

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

Step 4.1.5.10.2

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

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

Step 4.1.5.11

Add $6 h$ and $0$ .

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Factor $h$ out of $6 h$ .

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

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

Step 5