Calculus Examples

$w = (\frac{4 + 9 z}{3 z}) (6 - z)$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d z} [f (z) g (z)]$ is $f (z) \frac{d}{d z} [g (z)] + g (z) \frac{d}{d z} [f (z)]$ where $f (z) = \frac{4 + 9 z}{3 z}$ and $g (z) = 6 - z$ .

$\frac{4 + 9 z}{3 z} \frac{d}{d z} [6 - z] + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{4 + 9 z}{3 z} (\frac{d}{d z} [6] + \frac{d}{d z} [- z]) + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.2

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

$\frac{4 + 9 z}{3 z} (0 + \frac{d}{d z} [- z]) + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.3

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

$\frac{4 + 9 z}{3 z} \frac{d}{d z} [- z] + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.4

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

$\frac{4 + 9 z}{3 z} (- \frac{d}{d z} [z]) + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.5

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

$\frac{4 + 9 z}{3 z} (- 1 \cdot 1) + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.6

Combine fractions.

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Step 1.2.6.1

Multiply $- 1$ by $1$ .

$\frac{4 + 9 z}{3 z} \cdot - 1 + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.6.2

Combine $\frac{4 + 9 z}{3 z}$ and $- 1$ .

$\frac{(4 + 9 z) \cdot - 1}{3 z} + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.6.3

Simplify the expression.

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Step 1.2.6.3.1

Move $- 1$ to the left of $4 + 9 z$ .

$\frac{- 1 \cdot (4 + 9 z)}{3 z} + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.6.3.2

Rewrite $- 1 (4 + 9 z)$ as $- (4 + 9 z)$ .

$\frac{- (4 + 9 z)}{3 z} + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.6.3.3

Move the negative in front of the fraction.

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{d}{d z} [\frac{4 + 9 z}{3 z}]$

Step 1.2.7

Since $\frac{1}{3}$ is constant with respect to $z$ , the derivative of $\frac{4 + 9 z}{3 z}$ with respect to $z$ is $\frac{1}{3} \frac{d}{d z} [\frac{4 + 9 z}{z}]$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) (\frac{1}{3} \frac{d}{d z} [\frac{4 + 9 z}{z}])$

Step 1.3

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 $f (z) = 4 + 9 z$ and $g (z) = z$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{z \frac{d}{d z} [4 + 9 z] - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4

Differentiate.

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Step 1.4.1

By the Sum Rule, the derivative of $4 + 9 z$ with respect to $z$ is $\frac{d}{d z} [4] + \frac{d}{d z} [9 z]$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{z (\frac{d}{d z} [4] + \frac{d}{d z} [9 z]) - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.2

Since $4$ is constant with respect to $z$ , the derivative of $4$ with respect to $z$ is $0$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{z (0 + \frac{d}{d z} [9 z]) - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.3

Add $0$ and $\frac{d}{d z} [9 z]$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{z \frac{d}{d z} [9 z] - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.4

Since $9$ is constant with respect to $z$ , the derivative of $9 z$ with respect to $z$ is $9 \frac{d}{d z} [z]$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{z (9 \frac{d}{d z} [z]) - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.5

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

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{z (9 \cdot 1) - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.6

Simplify the expression.

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Step 1.4.6.1

Multiply $9$ by $1$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{z \cdot 9 - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.6.2

Move $9$ to the left of $z$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{9 \cdot z - (4 + 9 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.7

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

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{9 z - (4 + 9 z) \cdot 1}{z^{2}}$

Step 1.4.8

Combine fractions.

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Step 1.4.8.1

Multiply $- 1$ by $1$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{1}{3} \frac{9 z - (4 + 9 z)}{z^{2}}$

Step 1.4.8.2

Multiply $\frac{9 z - (4 + 9 z)}{z^{2}}$ by $\frac{1}{3}$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{9 z - (4 + 9 z)}{z^{2} \cdot 3}$

Step 1.4.8.3

Move $3$ to the left of $z^{2}$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{9 z - (4 + 9 z)}{3 \cdot z^{2}}$

Step 1.5

To write $(6 - z) \frac{9 z - (4 + 9 z)}{3 z^{2}}$ as a fraction with a common denominator, multiply by $\frac{3 z}{3 z}$ .

$- \frac{4 + 9 z}{3 z} + (6 - z) \frac{9 z - (4 + 9 z)}{3 z^{2}} \cdot \frac{3 z}{3 z}$

Step 1.6

Combine $(6 - z) \frac{9 z - (4 + 9 z)}{3 z^{2}}$ and $\frac{3 z}{3 z}$ .

$- \frac{4 + 9 z}{3 z} + \frac{(6 - z) \frac{9 z - (4 + 9 z)}{3 z^{2}} (3 z)}{3 z}$

Step 1.7

Combine the numerators over the common denominator.

$\frac{- (4 + 9 z) + (6 - z) \frac{9 z - (4 + 9 z)}{3 z^{2}} (3 z)}{3 z}$

Step 1.8

Combine $3$ and $\frac{9 z - (4 + 9 z)}{3 z^{2}}$ .

$\frac{- (4 + 9 z) + (6 - z) \frac{3 (9 z - (4 + 9 z))}{3 z^{2}} z}{3 z}$

Step 1.9

Combine $z$ and $\frac{3 (9 z - (4 + 9 z))}{3 z^{2}}$ .

$\frac{- (4 + 9 z) + (6 - z) \frac{z (3 (9 z - (4 + 9 z)))}{3 z^{2}}}{3 z}$

Step 1.10

Cancel the common factors.

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Step 1.10.1

Factor $z$ out of $3 z^{2}$ .

$\frac{- (4 + 9 z) + (6 - z) \frac{z (3 (9 z - (4 + 9 z)))}{z (3 z)}}{3 z}$

Step 1.10.2

Cancel the common factor.

$\frac{- (4 + 9 z) + (6 - z) \frac{z (3 (9 z - (4 + 9 z)))}{z (3 z)}}{3 z}$

Step 1.10.3

Rewrite the expression.

$\frac{- (4 + 9 z) + (6 - z) \frac{3 (9 z - (4 + 9 z))}{3 z}}{3 z}$

Step 1.11

Cancel the common factor of $3$ .

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Step 1.11.1

Cancel the common factor.

$\frac{- (4 + 9 z) + (6 - z) \frac{3 (9 z - (4 + 9 z))}{3 z}}{3 z}$

Step 1.11.2

Rewrite the expression.

$\frac{- (4 + 9 z) + (6 - z) \frac{9 z - (4 + 9 z)}{z}}{3 z}$

Step 1.12

Simplify.

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Step 1.12.1

Apply the distributive property.

$\frac{- 1 \cdot 4 - (9 z) + (6 - z) \frac{9 z - (4 + 9 z)}{z}}{3 z}$

Step 1.12.2

Apply the distributive property.

$\frac{- 1 \cdot 4 - (9 z) + (6 - z) \frac{9 z - 1 \cdot 4 - (9 z)}{z}}{3 z}$

Step 1.12.3

Simplify the numerator.

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Step 1.12.3.1

Simplify each term.

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Step 1.12.3.1.1

Multiply $- 1$ by $4$ .

$\frac{- 4 - (9 z) + (6 - z) \frac{9 z - 1 \cdot 4 - (9 z)}{z}}{3 z}$

Step 1.12.3.1.2

Multiply $9$ by $- 1$ .

$\frac{- 4 - 9 z + (6 - z) \frac{9 z - 1 \cdot 4 - (9 z)}{z}}{3 z}$

Step 1.12.3.1.3

Simplify the numerator.

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Step 1.12.3.1.3.1

Multiply $- 1$ by $4$ .

$\frac{- 4 - 9 z + (6 - z) \frac{9 z - 4 - (9 z)}{z}}{3 z}$

Step 1.12.3.1.3.2

Multiply $9$ by $- 1$ .

$\frac{- 4 - 9 z + (6 - z) \frac{9 z - 4 - 9 z}{z}}{3 z}$

Step 1.12.3.1.3.3

Subtract $9 z$ from $9 z$ .

$\frac{- 4 - 9 z + (6 - z) \frac{0 - 4}{z}}{3 z}$

Step 1.12.3.1.3.4

Subtract $4$ from $0$ .

$\frac{- 4 - 9 z + (6 - z) \frac{- 4}{z}}{3 z}$

Step 1.12.3.1.4

Move the negative in front of the fraction.

$\frac{- 4 - 9 z + (6 - z) (- \frac{4}{z})}{3 z}$

Step 1.12.3.1.5

Apply the distributive property.

$\frac{- 4 - 9 z + 6 (- \frac{4}{z}) - z (- \frac{4}{z})}{3 z}$

Step 1.12.3.1.6

Multiply $6 (- \frac{4}{z})$ .

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Step 1.12.3.1.6.1

Multiply $- 1$ by $6$ .

$\frac{- 4 - 9 z - 6 \frac{4}{z} - z (- \frac{4}{z})}{3 z}$

Step 1.12.3.1.6.2

Combine $- 6$ and $\frac{4}{z}$ .

$\frac{- 4 - 9 z + \frac{- 6 \cdot 4}{z} - z (- \frac{4}{z})}{3 z}$

Step 1.12.3.1.6.3

Multiply $- 6$ by $4$ .

$\frac{- 4 - 9 z + \frac{- 24}{z} - z (- \frac{4}{z})}{3 z}$

Step 1.12.3.1.7

Cancel the common factor of $z$ .

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Step 1.12.3.1.7.1

Move the leading negative in $- \frac{4}{z}$ into the numerator.

$\frac{- 4 - 9 z + \frac{- 24}{z} - z \frac{- 4}{z}}{3 z}$

Step 1.12.3.1.7.2

Factor $z$ out of $- z$ .

$\frac{- 4 - 9 z + \frac{- 24}{z} + z \cdot - 1 \frac{- 4}{z}}{3 z}$

Step 1.12.3.1.7.3

Cancel the common factor.

$\frac{- 4 - 9 z + \frac{- 24}{z} + z \cdot - 1 \frac{- 4}{z}}{3 z}$

Step 1.12.3.1.7.4

Rewrite the expression.

$\frac{- 4 - 9 z + \frac{- 24}{z} - 1 \cdot - 4}{3 z}$

Step 1.12.3.1.8

Multiply $- 1$ by $- 4$ .

$\frac{- 4 - 9 z + \frac{- 24}{z} + 4}{3 z}$

Step 1.12.3.1.9

Move the negative in front of the fraction.

$\frac{- 4 - 9 z - \frac{24}{z} + 4}{3 z}$

Step 1.12.3.2

Combine the opposite terms in $- 4 - 9 z - \frac{24}{z} + 4$ .

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Step 1.12.3.2.1

Add $- 4$ and $4$ .

$\frac{- 9 z - \frac{24}{z} + 0}{3 z}$

Step 1.12.3.2.2

Add $- 9 z - \frac{24}{z}$ and $0$ .

$\frac{- 9 z - \frac{24}{z}}{3 z}$

Step 1.12.4

Cancel the common factor of $- 9 z - \frac{24}{z}$ and $3$ .

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Step 1.12.4.1

Factor $3$ out of $- 9 z$ .

$\frac{3 (- 3 z) - \frac{24}{z}}{3 z}$

Step 1.12.4.2

Factor $3$ out of $- \frac{24}{z}$ .

$\frac{3 (- 3 z) + 3 (- \frac{8}{z})}{3 z}$

Step 1.12.4.3

Factor $3$ out of $3 (- 3 z) + 3 (- \frac{8}{z})$ .

$\frac{3 (- 3 z - \frac{8}{z})}{3 z}$

Step 1.12.4.4

Cancel the common factors.

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Step 1.12.4.4.1

Factor $3$ out of $3 z$ .

$\frac{3 (- 3 z - \frac{8}{z})}{3 (z)}$

Step 1.12.4.4.2

Cancel the common factor.

$\frac{3 (- 3 z - \frac{8}{z})}{3 z}$

Step 1.12.4.4.3

Rewrite the expression.

$\frac{- 3 z - \frac{8}{z}}{z}$

Step 1.12.5

Simplify the numerator.

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Step 1.12.5.1

To write $- 3 z$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{- 3 z \cdot \frac{z}{z} - \frac{8}{z}}{z}$

Step 1.12.5.2

Combine $- 3 z$ and $\frac{z}{z}$ .

$\frac{\frac{- 3 z \cdot z}{z} - \frac{8}{z}}{z}$

Step 1.12.5.3

Combine the numerators over the common denominator.

$\frac{\frac{- 3 z \cdot z - 8}{z}}{z}$

Step 1.12.5.4

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

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Step 1.12.5.4.1

Move $z$ .

$\frac{\frac{- 3 (z \cdot z) - 8}{z}}{z}$

Step 1.12.5.4.2

Multiply $z$ by $z$ .

$\frac{\frac{- 3 z^{2} - 8}{z}}{z}$

Step 1.12.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 3 z^{2} - 8}{z} \cdot \frac{1}{z}$

Step 1.12.7

Multiply $\frac{- 3 z^{2} - 8}{z} \cdot \frac{1}{z}$ .

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Step 1.12.7.1

Multiply $\frac{- 3 z^{2} - 8}{z}$ by $\frac{1}{z}$ .

$\frac{- 3 z^{2} - 8}{z \cdot z}$

Step 1.12.7.2

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

$\frac{- 3 z^{2} - 8}{z^{1} z}$

Step 1.12.7.3

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

$\frac{- 3 z^{2} - 8}{z^{1} z^{1}}$

Step 1.12.7.4

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

$\frac{- 3 z^{2} - 8}{z^{1 + 1}}$

Step 1.12.7.5

Add $1$ and $1$ .

$\frac{- 3 z^{2} - 8}{z^{2}}$

Step 1.12.8

Factor $- 1$ out of $- 3 z^{2}$ .

$\frac{- (3 z^{2}) - 8}{z^{2}}$

Step 1.12.9

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- (3 z^{2}) - 1 (8)}{z^{2}}$

Step 1.12.10

Factor $- 1$ out of $- (3 z^{2}) - 1 (8)$ .

$\frac{- (3 z^{2} + 8)}{z^{2}}$

Step 1.12.11

Rewrite $- (3 z^{2} + 8)$ as $- 1 (3 z^{2} + 8)$ .

$\frac{- 1 (3 z^{2} + 8)}{z^{2}}$

Step 1.12.12

Move the negative in front of the fraction.

$f' (z) = - \frac{3 z^{2} + 8}{z^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d z} [f (z) g (z)]$ is $f (z) \frac{d}{d z} [g (z)] + g (z) \frac{d}{d z} [f (z)]$ where $f (z) = - 1$ and $g (z) = \frac{3 z^{2} + 8}{z^{2}}$ .

$- \frac{d}{d z} [\frac{3 z^{2} + 8}{z^{2}}] + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.2

$- \frac{z^{2} \frac{d}{d z} [3 z^{2} + 8] - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{{(z^{2})}^{2}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply the exponents in ${(z^{2})}^{2}$ .

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Step 2.3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{z^{2} \frac{d}{d z} [3 z^{2} + 8] - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{2 \cdot 2}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3.1.2

Multiply $2$ by $2$ .

$- \frac{z^{2} \frac{d}{d z} [3 z^{2} + 8] - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3.2

By the Sum Rule, the derivative of $3 z^{2} + 8$ with respect to $z$ is $\frac{d}{d z} [3 z^{2}] + \frac{d}{d z} [8]$ .

$- \frac{z^{2} (\frac{d}{d z} [3 z^{2}] + \frac{d}{d z} [8]) - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3.3

Since $3$ is constant with respect to $z$ , the derivative of $3 z^{2}$ with respect to $z$ is $3 \frac{d}{d z} [z^{2}]$ .

$- \frac{z^{2} (3 \frac{d}{d z} [z^{2}] + \frac{d}{d z} [8]) - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3.4

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

$- \frac{z^{2} (3 (2 z) + \frac{d}{d z} [8]) - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3.5

Multiply $2$ by $3$ .

$- \frac{z^{2} (6 z + \frac{d}{d z} [8]) - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3.6

Since $8$ is constant with respect to $z$ , the derivative of $8$ with respect to $z$ is $0$ .

$- \frac{z^{2} (6 z + 0) - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.3.7

Add $6 z$ and $0$ .

$- \frac{z^{2} (6 z) - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.4

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

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Step 2.4.1

Move $z$ .

$- \frac{z \cdot z^{2} \cdot 6 - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.4.2

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

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Step 2.4.2.1

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

$- \frac{z^{1} z^{2} \cdot 6 - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.4.2.2

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

$- \frac{z^{1 + 2} \cdot 6 - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.4.3

Add $1$ and $2$ .

$- \frac{z^{3} \cdot 6 - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.5

Move $6$ to the left of $z^{3}$ .

$- \frac{6 \cdot z^{3} - (3 z^{2} + 8) \frac{d}{d z} [z^{2}]}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.6

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

$- \frac{6 z^{3} - (3 z^{2} + 8) (2 z)}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.7

Multiply $2$ by $- 1$ .

$- \frac{6 z^{3} - 2 (3 z^{2} + 8) z}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \frac{d}{d z} [- 1]$

Step 2.8

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

$- \frac{6 z^{3} - 2 (3 z^{2} + 8) z}{z^{4}} + \frac{3 z^{2} + 8}{z^{2}} \cdot 0$

Step 2.9

Simplify the expression.

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Step 2.9.1

Multiply $\frac{3 z^{2} + 8}{z^{2}}$ by $0$ .

$- \frac{6 z^{3} - 2 (3 z^{2} + 8) z}{z^{4}} + 0$

Step 2.9.2

Add $- \frac{6 z^{3} - 2 (3 z^{2} + 8) z}{z^{4}}$ and $0$ .

$- \frac{6 z^{3} - 2 (3 z^{2} + 8) z}{z^{4}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$- \frac{6 z^{3} + (- 2 (3 z^{2}) - 2 \cdot 8) z}{z^{4}}$

Step 2.10.2

Apply the distributive property.

$- \frac{6 z^{3} - 2 (3 z^{2}) z - 2 \cdot 8 z}{z^{4}}$

Step 2.10.3

Simplify the numerator.

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Step 2.10.3.1

Simplify each term.

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Step 2.10.3.1.1

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

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Step 2.10.3.1.1.1

Move $z$ .

$- \frac{6 z^{3} - 2 (3 (z \cdot z^{2})) - 2 \cdot 8 z}{z^{4}}$

Step 2.10.3.1.1.2

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

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Step 2.10.3.1.1.2.1

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

$- \frac{6 z^{3} - 2 (3 (z^{1} z^{2})) - 2 \cdot 8 z}{z^{4}}$

Step 2.10.3.1.1.2.2

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

$- \frac{6 z^{3} - 2 (3 z^{1 + 2}) - 2 \cdot 8 z}{z^{4}}$

Step 2.10.3.1.1.3

Add $1$ and $2$ .

$- \frac{6 z^{3} - 2 (3 z^{3}) - 2 \cdot 8 z}{z^{4}}$

Step 2.10.3.1.2

Multiply $3$ by $- 2$ .

$- \frac{6 z^{3} - 6 z^{3} - 2 \cdot 8 z}{z^{4}}$

Step 2.10.3.1.3

Multiply $- 2$ by $8$ .

$- \frac{6 z^{3} - 6 z^{3} - 16 z}{z^{4}}$

Step 2.10.3.2

Subtract $6 z^{3}$ from $6 z^{3}$ .

$- \frac{0 - 16 z}{z^{4}}$

Step 2.10.3.3

Subtract $16 z$ from $0$ .

$- \frac{- 16 z}{z^{4}}$

Step 2.10.4

Combine terms.

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Step 2.10.4.1

Cancel the common factor of $z$ and $z^{4}$ .

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Step 2.10.4.1.1

Factor $z$ out of $- 16 z$ .

$- \frac{z \cdot - 16}{z^{4}}$

Step 2.10.4.1.2

Cancel the common factors.

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Step 2.10.4.1.2.1

Factor $z$ out of $z^{4}$ .

$- \frac{z \cdot - 16}{z \cdot z^{3}}$

Step 2.10.4.1.2.2

Cancel the common factor.

$- \frac{z \cdot - 16}{z \cdot z^{3}}$

Step 2.10.4.1.2.3

Rewrite the expression.

$- \frac{- 16}{z^{3}}$

Step 2.10.4.2

Move the negative in front of the fraction.

$- - \frac{16}{z^{3}}$

Step 2.10.4.3

Multiply $- 1$ by $- 1$ .

$1 \frac{16}{z^{3}}$

Step 2.10.4.4

Multiply $\frac{16}{z^{3}}$ by $1$ .

$f'' (z) = \frac{16}{z^{3}}$