Calculus Examples

$w = (\frac{1 + 15 z}{5 z}) (15 - z)$

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 $f (z) = \frac{1 + 15 z}{5 z}$ and $g (z) = 15 - z$ .

$\frac{1 + 15 z}{5 z} \frac{d}{d z} [15 - z] + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $15 - z$ with respect to $z$ is $\frac{d}{d z} [15] + \frac{d}{d z} [- z]$ .

$\frac{1 + 15 z}{5 z} (\frac{d}{d z} [15] + \frac{d}{d z} [- z]) + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.2

Since $15$ is constant with respect to $z$ , the derivative of $15$ with respect to $z$ is $0$ .

$\frac{1 + 15 z}{5 z} (0 + \frac{d}{d z} [- z]) + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.3

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

$\frac{1 + 15 z}{5 z} \frac{d}{d z} [- z] + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.4

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

$\frac{1 + 15 z}{5 z} (- \frac{d}{d z} [z]) + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.5

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

$\frac{1 + 15 z}{5 z} (- 1 \cdot 1) + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.6

Combine fractions.

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Step 1.2.6.1

Multiply $- 1$ by $1$ .

$\frac{1 + 15 z}{5 z} \cdot - 1 + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.6.2

Combine $\frac{1 + 15 z}{5 z}$ and $- 1$ .

$\frac{(1 + 15 z) \cdot - 1}{5 z} + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.6.3

Simplify the expression.

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Step 1.2.6.3.1

Move $- 1$ to the left of $1 + 15 z$ .

$\frac{- 1 \cdot (1 + 15 z)}{5 z} + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.6.3.2

Rewrite $- 1 (1 + 15 z)$ as $- (1 + 15 z)$ .

$\frac{- (1 + 15 z)}{5 z} + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.6.3.3

Move the negative in front of the fraction.

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{d}{d z} [\frac{1 + 15 z}{5 z}]$

Step 1.2.7

Since $\frac{1}{5}$ is constant with respect to $z$ , the derivative of $\frac{1 + 15 z}{5 z}$ with respect to $z$ is $\frac{1}{5} \frac{d}{d z} [\frac{1 + 15 z}{z}]$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) (\frac{1}{5} \frac{d}{d z} [\frac{1 + 15 z}{z}])$

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 $f (z) = 1 + 15 z$ and $g (z) = z$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{z \frac{d}{d z} [1 + 15 z] - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4

Differentiate.

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Step 1.4.1

By the Sum Rule, the derivative of $1 + 15 z$ with respect to $z$ is $\frac{d}{d z} [1] + \frac{d}{d z} [15 z]$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{z (\frac{d}{d z} [1] + \frac{d}{d z} [15 z]) - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.2

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

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{z (0 + \frac{d}{d z} [15 z]) - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.3

Add $0$ and $\frac{d}{d z} [15 z]$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{z \frac{d}{d z} [15 z] - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.4

Since $15$ is constant with respect to $z$ , the derivative of $15 z$ with respect to $z$ is $15 \frac{d}{d z} [z]$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{z (15 \frac{d}{d z} [z]) - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.5

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

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{z (15 \cdot 1) - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.6

Simplify the expression.

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Step 1.4.6.1

Multiply $15$ by $1$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{z \cdot 15 - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.6.2

Move $15$ to the left of $z$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{15 \cdot z - (1 + 15 z) \frac{d}{d z} [z]}{z^{2}}$

Step 1.4.7

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

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{15 z - (1 + 15 z) \cdot 1}{z^{2}}$

Step 1.4.8

Combine fractions.

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Step 1.4.8.1

Multiply $- 1$ by $1$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{1}{5} \frac{15 z - (1 + 15 z)}{z^{2}}$

Step 1.4.8.2

Multiply $\frac{15 z - (1 + 15 z)}{z^{2}}$ by $\frac{1}{5}$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{15 z - (1 + 15 z)}{z^{2} \cdot 5}$

Step 1.4.8.3

Move $5$ to the left of $z^{2}$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{15 z - (1 + 15 z)}{5 \cdot z^{2}}$

Step 1.5

To write $(15 - z) \frac{15 z - (1 + 15 z)}{5 z^{2}}$ as a fraction with a common denominator, multiply by $\frac{5 z}{5 z}$ .

$- \frac{1 + 15 z}{5 z} + (15 - z) \frac{15 z - (1 + 15 z)}{5 z^{2}} \cdot \frac{5 z}{5 z}$

Step 1.6

Combine $(15 - z) \frac{15 z - (1 + 15 z)}{5 z^{2}}$ and $\frac{5 z}{5 z}$ .

$- \frac{1 + 15 z}{5 z} + \frac{(15 - z) \frac{15 z - (1 + 15 z)}{5 z^{2}} (5 z)}{5 z}$

Step 1.7

Combine the numerators over the common denominator.

$\frac{- (1 + 15 z) + (15 - z) \frac{15 z - (1 + 15 z)}{5 z^{2}} (5 z)}{5 z}$

Step 1.8

Combine $5$ and $\frac{15 z - (1 + 15 z)}{5 z^{2}}$ .

$\frac{- (1 + 15 z) + (15 - z) \frac{5 (15 z - (1 + 15 z))}{5 z^{2}} z}{5 z}$

Step 1.9

Combine $z$ and $\frac{5 (15 z - (1 + 15 z))}{5 z^{2}}$ .

$\frac{- (1 + 15 z) + (15 - z) \frac{z (5 (15 z - (1 + 15 z)))}{5 z^{2}}}{5 z}$

Step 1.10

Cancel the common factors.

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Step 1.10.1

Factor $z$ out of $5 z^{2}$ .

$\frac{- (1 + 15 z) + (15 - z) \frac{z (5 (15 z - (1 + 15 z)))}{z (5 z)}}{5 z}$

Step 1.10.2

Cancel the common factor.

$\frac{- (1 + 15 z) + (15 - z) \frac{z (5 (15 z - (1 + 15 z)))}{z (5 z)}}{5 z}$

Step 1.10.3

Rewrite the expression.

$\frac{- (1 + 15 z) + (15 - z) \frac{5 (15 z - (1 + 15 z))}{5 z}}{5 z}$

Step 1.11

Cancel the common factor of $5$ .

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Step 1.11.1

Cancel the common factor.

$\frac{- (1 + 15 z) + (15 - z) \frac{5 (15 z - (1 + 15 z))}{5 z}}{5 z}$

Step 1.11.2

Rewrite the expression.

$\frac{- (1 + 15 z) + (15 - z) \frac{15 z - (1 + 15 z)}{z}}{5 z}$

Step 1.12

Simplify.

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Step 1.12.1

Apply the distributive property.

$\frac{- 1 \cdot 1 - (15 z) + (15 - z) \frac{15 z - (1 + 15 z)}{z}}{5 z}$

Step 1.12.2

Apply the distributive property.

$\frac{- 1 \cdot 1 - (15 z) + (15 - z) \frac{15 z - 1 \cdot 1 - (15 z)}{z}}{5 z}$

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 $1$ .

$\frac{- 1 - (15 z) + (15 - z) \frac{15 z - 1 \cdot 1 - (15 z)}{z}}{5 z}$

Step 1.12.3.1.2

Multiply $15$ by $- 1$ .

$\frac{- 1 - 15 z + (15 - z) \frac{15 z - 1 \cdot 1 - (15 z)}{z}}{5 z}$

Step 1.12.3.1.3

Simplify the numerator.

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Step 1.12.3.1.3.1

Multiply $- 1$ by $1$ .

$\frac{- 1 - 15 z + (15 - z) \frac{15 z - 1 - (15 z)}{z}}{5 z}$

Step 1.12.3.1.3.2

Multiply $15$ by $- 1$ .

$\frac{- 1 - 15 z + (15 - z) \frac{15 z - 1 - 15 z}{z}}{5 z}$

Step 1.12.3.1.3.3

Subtract $15 z$ from $15 z$ .

$\frac{- 1 - 15 z + (15 - z) \frac{0 - 1}{z}}{5 z}$

Step 1.12.3.1.3.4

Subtract $1$ from $0$ .

$\frac{- 1 - 15 z + (15 - z) \frac{- 1}{z}}{5 z}$

Step 1.12.3.1.4

Move the negative in front of the fraction.

$\frac{- 1 - 15 z + (15 - z) (- \frac{1}{z})}{5 z}$

Step 1.12.3.1.5

Apply the distributive property.

$\frac{- 1 - 15 z + 15 (- \frac{1}{z}) - z (- \frac{1}{z})}{5 z}$

Step 1.12.3.1.6

Multiply $15 (- \frac{1}{z})$ .

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Step 1.12.3.1.6.1

Multiply $- 1$ by $15$ .

$\frac{- 1 - 15 z - 15 \frac{1}{z} - z (- \frac{1}{z})}{5 z}$

Step 1.12.3.1.6.2

Combine $- 15$ and $\frac{1}{z}$ .

$\frac{- 1 - 15 z + \frac{- 15}{z} - z (- \frac{1}{z})}{5 z}$

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 $- \frac{1}{z}$ into the numerator.

$\frac{- 1 - 15 z + \frac{- 15}{z} - z \frac{- 1}{z}}{5 z}$

Step 1.12.3.1.7.2

Factor $z$ out of $- z$ .

$\frac{- 1 - 15 z + \frac{- 15}{z} + z \cdot - 1 \frac{- 1}{z}}{5 z}$

Step 1.12.3.1.7.3

Cancel the common factor.

$\frac{- 1 - 15 z + \frac{- 15}{z} + z \cdot - 1 \frac{- 1}{z}}{5 z}$

Step 1.12.3.1.7.4

Rewrite the expression.

$\frac{- 1 - 15 z + \frac{- 15}{z} - 1 \cdot - 1}{5 z}$

Step 1.12.3.1.8

Multiply $- 1$ by $- 1$ .

$\frac{- 1 - 15 z + \frac{- 15}{z} + 1}{5 z}$

Step 1.12.3.1.9

Move the negative in front of the fraction.

$\frac{- 1 - 15 z - \frac{15}{z} + 1}{5 z}$

Step 1.12.3.2

Combine the opposite terms in $- 1 - 15 z - \frac{15}{z} + 1$ .

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Step 1.12.3.2.1

Add $- 1$ and $1$ .

$\frac{- 15 z - \frac{15}{z} + 0}{5 z}$

Step 1.12.3.2.2

Add $- 15 z - \frac{15}{z}$ and $0$ .

$\frac{- 15 z - \frac{15}{z}}{5 z}$

Step 1.12.4

Cancel the common factor of $- 15 z - \frac{15}{z}$ and $5$ .

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Step 1.12.4.1

Factor $5$ out of $- 15 z$ .

$\frac{5 (- 3 z) - \frac{15}{z}}{5 z}$

Step 1.12.4.2

Factor $5$ out of $- \frac{15}{z}$ .

$\frac{5 (- 3 z) + 5 (- \frac{3}{z})}{5 z}$

Step 1.12.4.3

Factor $5$ out of $5 (- 3 z) + 5 (- \frac{3}{z})$ .

$\frac{5 (- 3 z - \frac{3}{z})}{5 z}$

Step 1.12.4.4

Cancel the common factors.

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Step 1.12.4.4.1

Factor $5$ out of $5 z$ .

$\frac{5 (- 3 z - \frac{3}{z})}{5 (z)}$

Step 1.12.4.4.2

Cancel the common factor.

$\frac{5 (- 3 z - \frac{3}{z})}{5 z}$

Step 1.12.4.4.3

Rewrite the expression.

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

Step 1.12.5

Simplify the numerator.

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Step 1.12.5.1

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

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Step 1.12.5.1.1

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

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

Step 1.12.5.1.2

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

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

Step 1.12.5.1.3

Factor $3$ out of $3 (- z) + 3 (- \frac{1}{z})$ .

$\frac{3 (- z - \frac{1}{z})}{z}$

Step 1.12.5.2

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

$\frac{3 (- z \cdot \frac{z}{z} - \frac{1}{z})}{z}$

Step 1.12.5.3

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

$\frac{3 (\frac{- z \cdot z}{z} - \frac{1}{z})}{z}$

Step 1.12.5.4

Combine the numerators over the common denominator.

$\frac{3 \frac{- z \cdot z - 1}{z}}{z}$

Step 1.12.5.5

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

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Step 1.12.5.5.1

Move $z$ .

$\frac{3 \frac{- (z \cdot z) - 1}{z}}{z}$

Step 1.12.5.5.2

Multiply $z$ by $z$ .

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

Step 1.12.6

Combine $3$ and $\frac{- z^{2} - 1}{z}$ .

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

Step 1.12.7

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.12.8

Combine.

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

Step 1.12.9

Multiply $z$ by $z$ .

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

Step 1.12.10

Multiply $3 (- z^{2} - 1)$ by $1$ .

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

Step 1.12.11

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

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

Step 1.12.12

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

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

Step 1.12.13

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

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

Step 1.12.14

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

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

Step 1.12.15

Move the negative in front of the fraction.

$f' (z) = - \frac{3 (z^{2} + 1)}{z^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 3$ is constant with respect to $z$ , the derivative of $- \frac{3 (z^{2} + 1)}{z^{2}}$ with respect to $z$ is $- 3 \frac{d}{d z} [\frac{z^{2} + 1}{z^{2}}]$ .

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

Step 2.2

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.1.2

Multiply $2$ by $2$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $2 z$ and $0$ .

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

Step 2.4

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

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Step 2.4.1

Move $z$ .

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

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.3

Add $1$ and $2$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine fractions.

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Step 2.7.1

Multiply $2$ by $- 1$ .

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

Step 2.7.2

Combine $- 3$ and $\frac{2 z^{3} - 2 (z^{2} + 1) z}{z^{4}}$ .

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

Step 2.7.3

Move the negative in front of the fraction.

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

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Apply the distributive property.

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

Step 2.8.3

Apply the distributive property.

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

Step 2.8.4

Simplify the numerator.

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Step 2.8.4.1

Simplify each term.

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Step 2.8.4.1.1

Multiply $2$ by $3$ .

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

Step 2.8.4.1.2

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

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Step 2.8.4.1.2.1

Move $z$ .

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

Step 2.8.4.1.2.2

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

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Step 2.8.4.1.2.2.1

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

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

Step 2.8.4.1.2.2.2

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

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

Step 2.8.4.1.2.3

Add $1$ and $2$ .

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

Step 2.8.4.1.3

Multiply $- 2$ by $3$ .

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

Step 2.8.4.1.4

Multiply $- 2$ by $1$ .

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

Step 2.8.4.1.5

Multiply $- 2$ by $3$ .

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

Step 2.8.4.2

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

$- \frac{0 - 6 z}{z^{4}}$

Step 2.8.4.3

Subtract $6 z$ from $0$ .

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

Step 2.8.5

Combine terms.

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Step 2.8.5.1

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

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Step 2.8.5.1.1

Factor $z$ out of $- 6 z$ .

$- \frac{z \cdot - 6}{z^{4}}$

Step 2.8.5.1.2

Cancel the common factors.

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Step 2.8.5.1.2.1

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

$- \frac{z \cdot - 6}{z \cdot z^{3}}$

Step 2.8.5.1.2.2

Cancel the common factor.

$- \frac{z \cdot - 6}{z \cdot z^{3}}$

Step 2.8.5.1.2.3

Rewrite the expression.

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

Step 2.8.5.2

Move the negative in front of the fraction.

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

Step 2.8.5.3

Multiply $- 1$ by $- 1$ .

$1 \frac{6}{z^{3}}$

Step 2.8.5.4

Multiply $\frac{6}{z^{3}}$ by $1$ .

$f'' (z) = \frac{6}{z^{3}}$

Step 3

Find the third derivative.

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Step 3.1

Since $6$ is constant with respect to $z$ , the derivative of $\frac{6}{z^{3}}$ with respect to $z$ is $6 \frac{d}{d z} [\frac{1}{z^{3}}]$ .

$6 \frac{d}{d z} [\frac{1}{z^{3}}]$

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

Rewrite $\frac{1}{z^{3}}$ as ${(z^{3})}^{- 1}$ .

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

Step 3.2.2

Multiply the exponents in ${(z^{3})}^{- 1}$ .

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Step 3.2.2.1

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

$6 \frac{d}{d z} [z^{3 \cdot - 1}]$

Step 3.2.2.2

Multiply $3$ by $- 1$ .

$6 \frac{d}{d z} [z^{- 3}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = - 3$ .

$6 (- 3 z^{- 4})$

Step 3.4

Multiply $- 3$ by $6$ .

$- 18 z^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 18 \frac{1}{z^{4}}$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Combine $- 18$ and $\frac{1}{z^{4}}$ .

$\frac{- 18}{z^{4}}$

Step 3.5.2.2

Move the negative in front of the fraction.

$f''' (z) = - \frac{18}{z^{4}}$