Calculus Examples

$h (w) = \frac{6 w^{5} - w}{w}$

Step 1

Consider the difference quotient formula.

$\frac{f (w + h) - f w}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = w + h$ .

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Step 2.1.1

Replace the variable $w$ with $w + h$ in the expression.

$h (w + h) = \frac{6 {(w + h)}^{5} - (w + h)}{w + h}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Factor $w + h$ out of $6 {(w + h)}^{5} - (w + h)$ .

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Step 2.1.2.1.1.1

Factor $w + h$ out of $6 {(w + h)}^{5}$ .

$h (w + h) = \frac{(w + h) (6 {(w + h)}^{4}) - (w + h)}{w + h}$

Step 2.1.2.1.1.2

Factor $w + h$ out of $- (w + h)$ .

$h (w + h) = \frac{(w + h) (6 {(w + h)}^{4}) + (w + h) \cdot - 1}{w + h}$

Step 2.1.2.1.1.3

Factor $w + h$ out of $(w + h) (6 {(w + h)}^{4}) + (w + h) \cdot - 1$ .

$h (w + h) = \frac{(w + h) (6 {(w + h)}^{4} - 1)}{w + h}$

Step 2.1.2.1.2

Use the Binomial Theorem.

$h (w + h) = \frac{(w + h) (6 (w^{4} + 4 w^{3} h + 6 w^{2} h^{2} + 4 w h^{3} + h^{4}) - 1)}{w + h}$

Step 2.1.2.1.3

Apply the distributive property.

$h (w + h) = \frac{(w + h) (6 w^{4} + 6 (4 w^{3} h) + 6 (6 w^{2} h^{2}) + 6 (4 w h^{3}) + 6 h^{4} - 1)}{w + h}$

Step 2.1.2.1.4

Simplify.

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Step 2.1.2.1.4.1

Multiply $4$ by $6$ .

$h (w + h) = \frac{(w + h) (6 w^{4} + 24 (w^{3} h) + 6 (6 w^{2} h^{2}) + 6 (4 w h^{3}) + 6 h^{4} - 1)}{w + h}$

Step 2.1.2.1.4.2

Multiply $6$ by $6$ .

$h (w + h) = \frac{(w + h) (6 w^{4} + 24 (w^{3} h) + 36 (w^{2} h^{2}) + 6 (4 w h^{3}) + 6 h^{4} - 1)}{w + h}$

Step 2.1.2.1.4.3

Multiply $4$ by $6$ .

$h (w + h) = \frac{(w + h) (6 w^{4} + 24 (w^{3} h) + 36 (w^{2} h^{2}) + 24 (w h^{3}) + 6 h^{4} - 1)}{w + h}$

Step 2.1.2.1.5

Remove parentheses.

$h (w + h) = \frac{(w + h) (6 w^{4} + 24 w^{3} h + 36 w^{2} h^{2} + 24 w h^{3} + 6 h^{4} - 1)}{w + h}$

Step 2.1.2.2

Cancel the common factor of $w + h$ .

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Step 2.1.2.2.1

Cancel the common factor.

$h (w + h) = \frac{(w + h) (6 w^{4} + 24 w^{3} h + 36 w^{2} h^{2} + 24 w h^{3} + 6 h^{4} - 1)}{w + h}$

Step 2.1.2.2.2

Divide $6 w^{4} + 24 w^{3} h + 36 w^{2} h^{2} + 24 w h^{3} + 6 h^{4} - 1$ by $1$ .

$h (w + h) = 6 w^{4} + 24 w^{3} h + 36 w^{2} h^{2} + 24 w h^{3} + 6 h^{4} - 1$

Step 2.1.2.3

The final answer is $6 w^{4} + 24 w^{3} h + 36 w^{2} h^{2} + 24 w h^{3} + 6 h^{4} - 1$ .

$6 w^{4} + 24 w^{3} h + 36 w^{2} h^{2} + 24 w h^{3} + 6 h^{4} - 1$

Step 2.2

Reorder.

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Step 2.2.1

Move $w^{3}$ .

$6 w^{4} + 24 h w^{3} + 36 w^{2} h^{2} + 24 w h^{3} + 6 h^{4} - 1$

Step 2.2.2

Move $w^{2}$ .

$6 w^{4} + 24 h w^{3} + 36 h^{2} w^{2} + 24 w h^{3} + 6 h^{4} - 1$

Step 2.2.3

Move $w$ .

$6 w^{4} + 24 h w^{3} + 36 h^{2} w^{2} + 24 h^{3} w + 6 h^{4} - 1$

Step 2.2.4

Move $6 w^{4}$ .

$24 h w^{3} + 36 h^{2} w^{2} + 24 h^{3} w + 6 h^{4} + 6 w^{4} - 1$

Step 2.2.5

Move $24 h w^{3}$ .

$36 h^{2} w^{2} + 24 h^{3} w + 6 h^{4} + 24 h w^{3} + 6 w^{4} - 1$

Step 2.2.6

Move $36 h^{2} w^{2}$ .

$24 h^{3} w + 6 h^{4} + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1$

Step 2.2.7

Reorder $24 h^{3} w$ and $6 h^{4}$ .

$6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1$

Step 2.3

Find the components of the definition.

$h (w + h) = 6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1$

$h (w) = 6 w^{4} - 1$

$h (w + h) = 6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1$

$h (w) = 6 w^{4} - 1$

Step 3

Plug in the components.

$\frac{h (w + h) - h w}{h} = \frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1 - (6 w^{4} - 1)}{h}$

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.

$\frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1 - (6 w^{4}) - - 1}{h}$

Step 4.1.2

Multiply $6$ by $- 1$ .

$\frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1 - 6 w^{4} - - 1}{h}$

Step 4.1.3

Multiply $- 1$ by $- 1$ .

$\frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 6 w^{4} - 1 - 6 w^{4} + 1}{h}$

Step 4.1.4

Subtract $6 w^{4}$ from $6 w^{4}$ .

$\frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 0 - 1 + 1}{h}$

Step 4.1.5

Add $6 h^{4}$ and $0$ .

$\frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} - 1 + 1}{h}$

Step 4.1.6

Add $- 1$ and $1$ .

$\frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3} + 0}{h}$

Step 4.1.7

Add $6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3}$ and $0$ .

$\frac{6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3}}{h}$

Step 4.1.8

Factor $6 h$ out of $6 h^{4} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3}$ .

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Step 4.1.8.1

Factor $6 h$ out of $6 h^{4}$ .

$\frac{6 h \cdot h^{3} + 24 h^{3} w + 36 h^{2} w^{2} + 24 h w^{3}}{h}$

Step 4.1.8.2

Factor $6 h$ out of $24 h^{3} w$ .

$\frac{6 h \cdot h^{3} + 6 h (4 h^{2} w) + 36 h^{2} w^{2} + 24 h w^{3}}{h}$

Step 4.1.8.3

Factor $6 h$ out of $36 h^{2} w^{2}$ .

$\frac{6 h \cdot h^{3} + 6 h (4 h^{2} w) + 6 h (6 h w^{2}) + 24 h w^{3}}{h}$

Step 4.1.8.4

Factor $6 h$ out of $24 h w^{3}$ .

$\frac{6 h \cdot h^{3} + 6 h (4 h^{2} w) + 6 h (6 h w^{2}) + 6 h (4 w^{3})}{h}$

Step 4.1.8.5

Factor $6 h$ out of $6 h \cdot h^{3} + 6 h (4 h^{2} w)$ .

$\frac{6 h (h^{3} + 4 h^{2} w) + 6 h (6 h w^{2}) + 6 h (4 w^{3})}{h}$

Step 4.1.8.6

Factor $6 h$ out of $6 h (h^{3} + 4 h^{2} w) + 6 h (6 h w^{2})$ .

$\frac{6 h (h^{3} + 4 h^{2} w + 6 h w^{2}) + 6 h (4 w^{3})}{h}$

Step 4.1.8.7

Factor $6 h$ out of $6 h (h^{3} + 4 h^{2} w + 6 h w^{2}) + 6 h (4 w^{3})$ .

$\frac{6 h (h^{3} + 4 h^{2} w + 6 h w^{2} + 4 w^{3})}{h}$

Step 4.2

Simplify terms.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{6 h (h^{3} + 4 h^{2} w + 6 h w^{2} + 4 w^{3})}{h}$

Step 4.2.1.2

Divide $6 (h^{3} + 4 h^{2} w + 6 h w^{2} + 4 w^{3})$ by $1$ .

$6 (h^{3} + 4 h^{2} w + 6 h w^{2} + 4 w^{3})$

Step 4.2.2

Apply the distributive property.

$6 h^{3} + 6 (4 h^{2} w) + 6 (6 h w^{2}) + 6 (4 w^{3})$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $4$ by $6$ .

$6 h^{3} + 24 (h^{2} w) + 6 (6 h w^{2}) + 6 (4 w^{3})$

Step 4.3.2

Multiply $6$ by $6$ .

$6 h^{3} + 24 (h^{2} w) + 36 (h w^{2}) + 6 (4 w^{3})$

Step 4.3.3

Multiply $4$ by $6$ .

$6 h^{3} + 24 (h^{2} w) + 36 (h w^{2}) + 24 w^{3}$

Step 4.4

Remove parentheses.

$6 h^{3} + 24 h^{2} w + 36 h w^{2} + 24 w^{3}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Move $h^{2}$ .

$6 h^{3} + 24 w h^{2} + 36 h w^{2} + 24 w^{3}$

Step 4.5.2

Move $h$ .

$6 h^{3} + 24 w h^{2} + 36 w^{2} h + 24 w^{3}$

Step 4.5.3

Move $6 h^{3}$ .

$24 w h^{2} + 36 w^{2} h + 24 w^{3} + 6 h^{3}$

Step 4.5.4

Move $24 w h^{2}$ .

$36 w^{2} h + 24 w^{3} + 24 w h^{2} + 6 h^{3}$

Step 4.5.5

Reorder $36 w^{2} h$ and $24 w^{3}$ .

$24 w^{3} + 36 w^{2} h + 24 w h^{2} + 6 h^{3}$

Step 5