Calculus Examples

$\frac{w^{6} - w}{w}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d w} [\frac{f (w)}{g (w)}]$ is $\frac{g (w) \frac{d}{d w} [f (w)] - f (w) \frac{d}{d w} [g (w)]}{{g (w)}^{2}}$ where $f (w) = w^{6} - w$ and $g (w) = w$ .

$\frac{w \frac{d}{d w} [w^{6} - w] - (w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $w^{6} - w$ with respect to $w$ is $\frac{d}{d w} [w^{6}] + \frac{d}{d w} [- w]$ .

$\frac{w (\frac{d}{d w} [w^{6}] + \frac{d}{d w} [- w]) - (w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 6$ .

$\frac{w (6 w^{5} + \frac{d}{d w} [- w]) - (w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 3

Evaluate $\frac{d}{d w} [- w]$ .

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Step 3.1

Since $- 1$ is constant with respect to $w$ , the derivative of $- w$ with respect to $w$ is $- \frac{d}{d w} [w]$ .

$\frac{w (6 w^{5} - \frac{d}{d w} [w]) - (w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 1$ .

$\frac{w (6 w^{5} - 1 \cdot 1) - (w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 3.3

Multiply $- 1$ by $1$ .

$\frac{w (6 w^{5} - 1) - (w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 1$ .

$\frac{w (6 w^{5} - 1) - (w^{6} - w) \cdot 1}{w^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{w (6 w^{5}) + w \cdot - 1 - (w^{6} - w) \cdot 1}{w^{2}}$

Step 5.2

Apply the distributive property.

$\frac{w (6 w^{5}) + w \cdot - 1 + (- w^{6} - - w) \cdot 1}{w^{2}}$

Step 5.3

Apply the distributive property.

$\frac{w (6 w^{5}) + w \cdot - 1 - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{6 w \cdot w^{5} + w \cdot - 1 - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4.1.2

Multiply $w$ by $w^{5}$ by adding the exponents.

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Step 5.4.1.2.1

Move $w^{5}$ .

$\frac{6 (w^{5} w) + w \cdot - 1 - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4.1.2.2

Multiply $w^{5}$ by $w$ .

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Step 5.4.1.2.2.1

Raise $w$ to the power of $1$ .

$\frac{6 (w^{5} w^{1}) + w \cdot - 1 - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 w^{5 + 1} + w \cdot - 1 - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4.1.2.3

Add $5$ and $1$ .

$\frac{6 w^{6} + w \cdot - 1 - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4.1.3

Move $- 1$ to the left of $w$ .

$\frac{6 w^{6} - 1 \cdot w - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4.1.4

Rewrite $- 1 w$ as $- w$ .

$\frac{6 w^{6} - w - w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 5.4.1.5

Multiply $- 1$ by $1$ .

$\frac{6 w^{6} - w - w^{6} - - w \cdot 1}{w^{2}}$

Step 5.4.1.6

Multiply $- - w$ .

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Step 5.4.1.6.1

Multiply $- 1$ by $- 1$ .

$\frac{6 w^{6} - w - w^{6} + 1 w \cdot 1}{w^{2}}$

Step 5.4.1.6.2

Multiply $w$ by $1$ .

$\frac{6 w^{6} - w - w^{6} + w \cdot 1}{w^{2}}$

Step 5.4.1.7

Multiply $w$ by $1$ .

$\frac{6 w^{6} - w - w^{6} + w}{w^{2}}$

Step 5.4.2

Combine the opposite terms in $6 w^{6} - w - w^{6} + w$ .

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Step 5.4.2.1

Add $- w$ and $w$ .

$\frac{6 w^{6} - w^{6} + 0}{w^{2}}$

Step 5.4.2.2

Add $6 w^{6} - w^{6}$ and $0$ .

$\frac{6 w^{6} - w^{6}}{w^{2}}$

Step 5.4.3

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

$\frac{5 w^{6}}{w^{2}}$

Step 5.5

Cancel the common factor of $w^{6}$ and $w^{2}$ .

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Step 5.5.1

Factor $w^{2}$ out of $5 w^{6}$ .

$\frac{w^{2} (5 w^{4})}{w^{2}}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Multiply by $1$ .

$\frac{w^{2} (5 w^{4})}{w^{2} \cdot 1}$

Step 5.5.2.2

Cancel the common factor.

$\frac{w^{2} (5 w^{4})}{w^{2} \cdot 1}$

Step 5.5.2.3

Rewrite the expression.

$\frac{5 w^{4}}{1}$

Step 5.5.2.4

Divide $5 w^{4}$ by $1$ .

$5 w^{4}$