Calculus Examples

$h (z) = \frac{1 - 2 z}{z^{2} - z - 6}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d z} [\frac{f (z)}{g (z)}]$ is $\frac{g (z) \frac{d}{d z} [f (z)] - f (z) \frac{d}{d z} [g (z)]}{{g (z)}^{2}}$ where $f (z) = 1 - 2 z$ and $g (z) = z^{2} - z - 6$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d z} [- 2 z]$ .

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

Step 2.4

Since $- 2$ is constant with respect to $z$ , the derivative of $- 2 z$ with respect to $z$ is $- 2 \frac{d}{d z} [z]$ .

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

Step 2.5

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

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

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 2$ by $1$ .

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

Step 2.6.2

Move $- 2$ to the left of $z^{2} - z - 6$ .

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

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Multiply $- 1$ by $1$ .

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

Step 2.12

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

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

Step 2.13

Simplify with factoring out.

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Step 2.13.1

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

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

Step 2.13.2

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

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

Step 2.13.3

Factor $- 1$ out of $- 2 z$ .

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

Step 2.13.4

Factor $- 1$ out of $- 1 (- 1) - (2 z)$ .

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

Step 2.13.5

Reorder terms.

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

Step 3

Raise $2 z - 1$ to the power of $1$ .

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

Step 4

Raise $2 z - 1$ to the power of $1$ .

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

Multiply $- 1$ by $- 1$ .

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

Step 8

Multiply ${(2 z - 1)}^{2}$ by $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Simplify the numerator.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Multiply $- 1$ by $- 2$ .

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

Step 9.2.1.2

Multiply $- 2$ by $- 6$ .

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

Step 9.2.1.3

Rewrite ${(2 z - 1)}^{2}$ as $(2 z - 1) (2 z - 1)$ .

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

Step 9.2.1.4

Expand $(2 z - 1) (2 z - 1)$ using the FOIL Method.

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Step 9.2.1.4.1

Apply the distributive property.

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

Step 9.2.1.4.2

Apply the distributive property.

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

Step 9.2.1.4.3

Apply the distributive property.

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

Step 9.2.1.5

Simplify and combine like terms.

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Step 9.2.1.5.1

Simplify each term.

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Step 9.2.1.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 9.2.1.5.1.2

Multiply $z$ by $z$ by adding the exponents.

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Step 9.2.1.5.1.2.1

Move $z$ .

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

Step 9.2.1.5.1.2.2

Multiply $z$ by $z$ .

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

Step 9.2.1.5.1.3

Multiply $2$ by $2$ .

$\frac{- 2 z^{2} + 2 z + 12 + 4 z^{2} + 2 z \cdot - 1 - 1 (2 z) - 1 \cdot - 1}{{(z^{2} - z - 6)}^{2}}$

Step 9.2.1.5.1.4

Multiply $- 1$ by $2$ .

$\frac{- 2 z^{2} + 2 z + 12 + 4 z^{2} - 2 z - 1 (2 z) - 1 \cdot - 1}{{(z^{2} - z - 6)}^{2}}$

Step 9.2.1.5.1.5

Multiply $2$ by $- 1$ .

$\frac{- 2 z^{2} + 2 z + 12 + 4 z^{2} - 2 z - 2 z - 1 \cdot - 1}{{(z^{2} - z - 6)}^{2}}$

Step 9.2.1.5.1.6

Multiply $- 1$ by $- 1$ .

$\frac{- 2 z^{2} + 2 z + 12 + 4 z^{2} - 2 z - 2 z + 1}{{(z^{2} - z - 6)}^{2}}$

Step 9.2.1.5.2

Subtract $2 z$ from $- 2 z$ .

$\frac{- 2 z^{2} + 2 z + 12 + 4 z^{2} - 4 z + 1}{{(z^{2} - z - 6)}^{2}}$

Step 9.2.2

Add $- 2 z^{2}$ and $4 z^{2}$ .

$\frac{2 z^{2} + 2 z + 12 - 4 z + 1}{{(z^{2} - z - 6)}^{2}}$

Step 9.2.3

Subtract $4 z$ from $2 z$ .

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

Step 9.2.4

Add $12$ and $1$ .

$\frac{2 z^{2} - 2 z + 13}{{(z^{2} - z - 6)}^{2}}$

Step 9.3

Simplify the denominator.

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Step 9.3.1

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

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Step 9.3.1.1

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 9.3.1.2

Write the factored form using these integers.

$\frac{2 z^{2} - 2 z + 13}{{((z - 3) (z + 2))}^{2}}$

Step 9.3.2

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

$\frac{2 z^{2} - 2 z + 13}{{(z - 3)}^{2} {(z + 2)}^{2}}$