Calculus Examples

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

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

Step 1

Differentiate using the Quotient Rule which states that

\frac{d}{d w} [\frac{f (w)}{g (w)}]

$\frac{d}{d w} [\frac{f (w)}{g (w)}]$ is

\frac{g (w) \frac{d}{d w} [f (w)] - f (w) \frac{d}{d w} [g (w)]}{{g (w)}^{2}}

$\frac{g (w) \frac{d}{d w} [f (w)] - f (w) \frac{d}{d w} [g (w)]}{{g (w)}^{2}}$ where

f (w) = 5 w^{6} - w

$f (w) = 5 w^{6} - w$ and

g (w) = w

$g (w) = w$ .

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

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

Step 2

By the Sum Rule, the derivative of

5 w^{6} - w

$5 w^{6} - w$ with respect to

w

$w$ is

\frac{d}{d w} [5 w^{6}] + \frac{d}{d w} [- w]

$\frac{d}{d w} [5 w^{6}] + \frac{d}{d w} [- w]$ .

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

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

Step 3

Evaluate

\frac{d}{d w} [5 w^{6}]

$\frac{d}{d w} [5 w^{6}]$ .

Tap for more steps...

Step 3.1

Since

5

$5$ is constant with respect to

w

$w$ , the derivative of

5 w^{6}

$5 w^{6}$ with respect to

w

$w$ is

5 \frac{d}{d w} [w^{6}]

$5 \frac{d}{d w} [w^{6}]$ .

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

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

Step 3.2

Differentiate using the Power Rule which states that

$\frac{d}{d w} [w^{n}]$ is

$n w^{n - 1}$ where

$n = 6$ .

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

Step 3.3

Multiply

$6$ by

$5$ .

$\frac{w (30 w^{5} + \frac{d}{d w} [- w]) - (5 w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 4

Evaluate

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

Tap for more steps...

Step 4.1

Since

$- 1$ is constant with respect to

$w$ , the derivative of

$- w$ with respect to

$w$ is

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

$\frac{w (30 w^{5} - \frac{d}{d w} [w]) - (5 w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 4.2

Differentiate using the Power Rule which states that

$\frac{d}{d w} [w^{n}]$ is

$n w^{n - 1}$ where

$n = 1$ .

$\frac{w (30 w^{5} - 1 \cdot 1) - (5 w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 4.3

Multiply

$- 1$ by

$1$ .

$\frac{w (30 w^{5} - 1) - (5 w^{6} - w) \frac{d}{d w} [w]}{w^{2}}$

Step 5

Differentiate using the Power Rule which states that

$\frac{d}{d w} [w^{n}]$ is

$n w^{n - 1}$ where

$n = 1$ .

$\frac{w (30 w^{5} - 1) - (5 w^{6} - w) \cdot 1}{w^{2}}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{w (30 w^{5}) + w \cdot - 1 - (5 w^{6} - w) \cdot 1}{w^{2}}$

Step 6.2

Apply the distributive property.

$\frac{w (30 w^{5}) + w \cdot - 1 + (- (5 w^{6}) - - w) \cdot 1}{w^{2}}$

Step 6.3

Apply the distributive property.

$\frac{w (30 w^{5}) + w \cdot - 1 - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4

Simplify the numerator.

Tap for more steps...

Step 6.4.1

Simplify each term.

Tap for more steps...

Step 6.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{30 w \cdot w^{5} + w \cdot - 1 - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.2

Multiply

$w$ by

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

Tap for more steps...

Step 6.4.1.2.1

Move

$w^{5}$ .

$\frac{30 (w^{5} w) + w \cdot - 1 - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.2.2

Multiply

$w^{5}$ by

$w$ .

Tap for more steps...

Step 6.4.1.2.2.1

Raise

$w$ to the power of

$1$ .

$\frac{30 (w^{5} w^{1}) + w \cdot - 1 - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{30 w^{5 + 1} + w \cdot - 1 - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.2.3

Add

$5$ and

$1$ .

$\frac{30 w^{6} + w \cdot - 1 - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.3

Move

$- 1$ to the left of

$w$ .

$\frac{30 w^{6} - 1 \cdot w - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.4

Rewrite

$- 1 w$ as

$- w$ .

$\frac{30 w^{6} - w - (5 w^{6}) \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.5

Multiply

$5$ by

$- 1$ .

$\frac{30 w^{6} - w - 5 w^{6} \cdot 1 - - w \cdot 1}{w^{2}}$

Step 6.4.1.6

Multiply

$- 5$ by

$1$ .

$\frac{30 w^{6} - w - 5 w^{6} - - w \cdot 1}{w^{2}}$

Step 6.4.1.7

Multiply

$- - w$ .

Tap for more steps...

Step 6.4.1.7.1

Multiply

$- 1$ by

$- 1$ .

$\frac{30 w^{6} - w - 5 w^{6} + 1 w \cdot 1}{w^{2}}$

Step 6.4.1.7.2

Multiply

$w$ by

$1$ .

$\frac{30 w^{6} - w - 5 w^{6} + w \cdot 1}{w^{2}}$

Step 6.4.1.8

Multiply

$w$ by

$1$ .

$\frac{30 w^{6} - w - 5 w^{6} + w}{w^{2}}$

Step 6.4.2

Combine the opposite terms in

$30 w^{6} - w - 5 w^{6} + w$ .

Tap for more steps...

Step 6.4.2.1

Add

$- w$ and

$w$ .

$\frac{30 w^{6} - 5 w^{6} + 0}{w^{2}}$

Step 6.4.2.2

Add

$30 w^{6} - 5 w^{6}$ and

$0$ .

$\frac{30 w^{6} - 5 w^{6}}{w^{2}}$

Step 6.4.3

Subtract

$5 w^{6}$ from

$30 w^{6}$ .

$\frac{25 w^{6}}{w^{2}}$

Step 6.5

Cancel the common factor of

$w^{6}$ and

$w^{2}$ .

Tap for more steps...

Step 6.5.1

Factor

$w^{2}$ out of

$25 w^{6}$ .

$\frac{w^{2} (25 w^{4})}{w^{2}}$

Step 6.5.2

Cancel the common factors.

Tap for more steps...

Step 6.5.2.1

Multiply by

$1$ .

$\frac{w^{2} (25 w^{4})}{w^{2} \cdot 1}$

Step 6.5.2.2

Cancel the common factor.

$\frac{w^{2} (25 w^{4})}{w^{2} \cdot 1}$

Step 6.5.2.3

Rewrite the expression.

$\frac{25 w^{4}}{1}$

Step 6.5.2.4

Divide

$25 w^{4}$ by

$1$ .

$25 w^{4}$