Calculus Examples

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

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 $f (w) = w^{2} + 1$ and $g (w) = w^{2} - w - 6$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $w^{2} + 1$ with respect to $w$ is $\frac{d}{d w} [w^{2}] + \frac{d}{d w} [1]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 2$ .

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

Step 2.3

Since $1$ is constant with respect to $w$ , the derivative of $1$ with respect to $w$ is $0$ .

$\frac{(w^{2} - w - 6) (2 w + 0) - (w^{2} + 1) \frac{d}{d w} [w^{2} - w - 6]}{{(w^{2} - w - 6)}^{2}}$

Step 2.4

Add $2 w$ and $0$ .

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

Step 2.5

By the Sum Rule, the derivative of $w^{2} - w - 6$ with respect to $w$ is $\frac{d}{d w} [w^{2}] + \frac{d}{d w} [- w] + \frac{d}{d w} [- 6]$ .

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

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d w} [w^{n}]$ is $n w^{n - 1}$ where $n = 2$ .

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 4

Differentiate using the Constant Rule.

Tap for more steps...

Step 4.1

Since $- 6$ is constant with respect to $w$ , the derivative of $- 6$ with respect to $w$ is $0$ .

$\frac{(w^{2} - w - 6) (2 w) - (w^{2} + 1) (2 w - 1 + 0)}{{(w^{2} - w - 6)}^{2}}$

Step 4.2

Add $2 w - 1$ and $0$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Simplify the numerator.

Tap for more steps...

Step 5.3.1

Simplify each term.

Tap for more steps...

Step 5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.2

Multiply $w^{2}$ by $w$ by adding the exponents.

Tap for more steps...

Step 5.3.1.2.1

Move $w$ .

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

Step 5.3.1.2.2

Multiply $w$ by $w^{2}$ .

Tap for more steps...

Step 5.3.1.2.2.1

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

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

Step 5.3.1.2.2.2

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

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

Step 5.3.1.2.3

Add $1$ and $2$ .

$\frac{2 w^{3} - w (2 w) - 6 (2 w) + (- w^{2} - 1 \cdot 1) (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 w^{3} - 1 \cdot 2 w \cdot w - 6 (2 w) + (- w^{2} - 1 \cdot 1) (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.4

Multiply $w$ by $w$ by adding the exponents.

Tap for more steps...

Step 5.3.1.4.1

Move $w$ .

$\frac{2 w^{3} - 1 \cdot 2 (w \cdot w) - 6 (2 w) + (- w^{2} - 1 \cdot 1) (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.4.2

Multiply $w$ by $w$ .

$\frac{2 w^{3} - 1 \cdot 2 w^{2} - 6 (2 w) + (- w^{2} - 1 \cdot 1) (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.5

Multiply $- 1$ by $2$ .

$\frac{2 w^{3} - 2 w^{2} - 6 (2 w) + (- w^{2} - 1 \cdot 1) (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.6

Multiply $2$ by $- 6$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w + (- w^{2} - 1 \cdot 1) (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.7

Multiply $- 1$ by $1$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w + (- w^{2} - 1) (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.8

Expand $(- w^{2} - 1) (2 w - 1)$ using the FOIL Method.

Tap for more steps...

Step 5.3.1.8.1

Apply the distributive property.

$\frac{2 w^{3} - 2 w^{2} - 12 w - w^{2} (2 w - 1) - 1 (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.8.2

Apply the distributive property.

$\frac{2 w^{3} - 2 w^{2} - 12 w - w^{2} (2 w) - w^{2} \cdot - 1 - 1 (2 w - 1)}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.8.3

Apply the distributive property.

$\frac{2 w^{3} - 2 w^{2} - 12 w - w^{2} (2 w) - w^{2} \cdot - 1 - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9

Simplify each term.

Tap for more steps...

Step 5.3.1.9.1

Rewrite using the commutative property of multiplication.

$\frac{2 w^{3} - 2 w^{2} - 12 w - 1 \cdot 2 w^{2} w - w^{2} \cdot - 1 - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.2

Multiply $w^{2}$ by $w$ by adding the exponents.

Tap for more steps...

Step 5.3.1.9.2.1

Move $w$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w - 1 \cdot 2 (w \cdot w^{2}) - w^{2} \cdot - 1 - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.2.2

Multiply $w$ by $w^{2}$ .

Tap for more steps...

Step 5.3.1.9.2.2.1

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

$\frac{2 w^{3} - 2 w^{2} - 12 w - 1 \cdot 2 (w^{1} w^{2}) - w^{2} \cdot - 1 - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.2.2.2

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

$\frac{2 w^{3} - 2 w^{2} - 12 w - 1 \cdot 2 w^{1 + 2} - w^{2} \cdot - 1 - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.2.3

Add $1$ and $2$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w - 1 \cdot 2 w^{3} - w^{2} \cdot - 1 - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.3

Multiply $- 1$ by $2$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w - 2 w^{3} - w^{2} \cdot - 1 - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.4

Multiply $- w^{2} \cdot - 1$ .

Tap for more steps...

Step 5.3.1.9.4.1

Multiply $- 1$ by $- 1$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w - 2 w^{3} + 1 w^{2} - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.4.2

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

$\frac{2 w^{3} - 2 w^{2} - 12 w - 2 w^{3} + w^{2} - 1 (2 w) - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.5

Multiply $2$ by $- 1$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w - 2 w^{3} + w^{2} - 2 w - 1 \cdot - 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.1.9.6

Multiply $- 1$ by $- 1$ .

$\frac{2 w^{3} - 2 w^{2} - 12 w - 2 w^{3} + w^{2} - 2 w + 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.2

Combine the opposite terms in $2 w^{3} - 2 w^{2} - 12 w - 2 w^{3} + w^{2} - 2 w + 1$ .

Tap for more steps...

Step 5.3.2.1

Subtract $2 w^{3}$ from $2 w^{3}$ .

$\frac{- 2 w^{2} - 12 w + 0 + w^{2} - 2 w + 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.2.2

Add $- 2 w^{2} - 12 w$ and $0$ .

$\frac{- 2 w^{2} - 12 w + w^{2} - 2 w + 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.3

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

$\frac{- w^{2} - 12 w - 2 w + 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.3.4

Subtract $2 w$ from $- 12 w$ .

$\frac{- w^{2} - 14 w + 1}{{(w^{2} - w - 6)}^{2}}$

Step 5.4

Simplify the denominator.

Tap for more steps...

Step 5.4.1

Factor $w^{2} - w - 6$ using the AC method.

Tap for more steps...

Step 5.4.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 5.4.1.2

Write the factored form using these integers.

$\frac{- w^{2} - 14 w + 1}{{((w - 3) (w + 2))}^{2}}$

Step 5.4.2

Apply the product rule to $(w - 3) (w + 2)$ .

$\frac{- w^{2} - 14 w + 1}{{(w - 3)}^{2} {(w + 2)}^{2}}$

Step 5.5

Factor $- 1$ out of $- w^{2}$ .

$\frac{- (w^{2}) - 14 w + 1}{{(w - 3)}^{2} {(w + 2)}^{2}}$

Step 5.6

Factor $- 1$ out of $- 14 w$ .

$\frac{- (w^{2}) - (14 w) + 1}{{(w - 3)}^{2} {(w + 2)}^{2}}$

Step 5.7

Factor $- 1$ out of $- (w^{2}) - (14 w)$ .

$\frac{- (w^{2} + 14 w) + 1}{{(w - 3)}^{2} {(w + 2)}^{2}}$

Step 5.8

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (w^{2} + 14 w) - 1 (- 1)}{{(w - 3)}^{2} {(w + 2)}^{2}}$

Step 5.9

Factor $- 1$ out of $- (w^{2} + 14 w) - 1 (- 1)$ .

$\frac{- (w^{2} + 14 w - 1)}{{(w - 3)}^{2} {(w + 2)}^{2}}$

Step 5.10

Rewrite $- (w^{2} + 14 w - 1)$ as $- 1 (w^{2} + 14 w - 1)$ .

$\frac{- 1 (w^{2} + 14 w - 1)}{{(w - 3)}^{2} {(w + 2)}^{2}}$

Step 5.11

Move the negative in front of the fraction.

$- \frac{w^{2} + 14 w - 1}{{(w - 3)}^{2} {(w + 2)}^{2}}$