Calculus Examples

$h (z) = (3 - z) (z^{3} - 2 z + 4)$

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 $f (z) = 3 - z$ and $g (z) = z^{3} - 2 z + 4$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $z^{3} - 2 z + 4$ with respect to $z$ is $\frac{d}{d z} [z^{3}] + \frac{d}{d z} [- 2 z] + \frac{d}{d z} [4]$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $- 2$ by $1$ .

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

Step 1.2.6

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

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

Step 1.2.7

Add $3 z^{2} - 2$ and $0$ .

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

Step 1.2.8

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

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

Step 1.2.9

Since $3$ is constant with respect to $z$ , the derivative of $3$ with respect to $z$ is $0$ .

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

Step 1.2.10

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

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

Step 1.2.11

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

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

Step 1.2.12

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

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

Step 1.2.13

Simplify the expression.

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Step 1.2.13.1

Multiply $- 1$ by $1$ .

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

Step 1.2.13.2

Move $- 1$ to the left of $z^{3} - 2 z + 4$ .

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

Step 1.2.13.3

Rewrite $- 1 (z^{3} - 2 z + 4)$ as $- (z^{3} - 2 z + 4)$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.3.4

Apply the distributive property.

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

Step 1.3.5

Combine terms.

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Step 1.3.5.1

Multiply $3$ by $3$ .

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

Step 1.3.5.2

Multiply $3$ by $- 1$ .

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

Step 1.3.5.3

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

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Step 1.3.5.3.1

Move $z^{2}$ .

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

Step 1.3.5.3.2

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

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Step 1.3.5.3.2.1

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

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

Step 1.3.5.3.2.2

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

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

Step 1.3.5.3.3

Add $2$ and $1$ .

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

Step 1.3.5.4

Multiply $3$ by $- 2$ .

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

Step 1.3.5.5

Multiply $- 2$ by $- 1$ .

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

Step 1.3.5.6

Multiply $- 2$ by $- 1$ .

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

Step 1.3.5.7

Multiply $- 1$ by $4$ .

$9 z^{2} - 3 z^{3} - 6 + 2 z - z^{3} + 2 z - 4$

Step 1.3.5.8

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

$9 z^{2} - 4 z^{3} - 6 + 2 z + 2 z - 4$

Step 1.3.5.9

Add $2 z$ and $2 z$ .

$9 z^{2} - 4 z^{3} - 6 + 4 z - 4$

Step 1.3.5.10

Subtract $4$ from $- 6$ .

$9 z^{2} - 4 z^{3} + 4 z - 10$

Step 1.3.6

Reorder terms.

$f' (z) = - 4 z^{3} + 9 z^{2} + 4 z - 10$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 4 z^{3} + 9 z^{2} + 4 z - 10$ with respect to $z$ is $\frac{d}{d z} [- 4 z^{3}] + \frac{d}{d z} [9 z^{2}] + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$ .

$\frac{d}{d z} [- 4 z^{3}] + \frac{d}{d z} [9 z^{2}] + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$

Step 2.2

Evaluate $\frac{d}{d z} [- 4 z^{3}]$ .

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Step 2.2.1

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

$- 4 \frac{d}{d z} [z^{3}] + \frac{d}{d z} [9 z^{2}] + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$

Step 2.2.2

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

$- 4 (3 z^{2}) + \frac{d}{d z} [9 z^{2}] + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$

Step 2.2.3

Multiply $3$ by $- 4$ .

$- 12 z^{2} + \frac{d}{d z} [9 z^{2}] + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$

Step 2.3

Evaluate $\frac{d}{d z} [9 z^{2}]$ .

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Step 2.3.1

Since $9$ is constant with respect to $z$ , the derivative of $9 z^{2}$ with respect to $z$ is $9 \frac{d}{d z} [z^{2}]$ .

$- 12 z^{2} + 9 \frac{d}{d z} [z^{2}] + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$

Step 2.3.2

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

$- 12 z^{2} + 9 (2 z) + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$

Step 2.3.3

Multiply $2$ by $9$ .

$- 12 z^{2} + 18 z + \frac{d}{d z} [4 z] + \frac{d}{d z} [- 10]$

Step 2.4

Evaluate $\frac{d}{d z} [4 z]$ .

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Step 2.4.1

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

$- 12 z^{2} + 18 z + 4 \frac{d}{d z} [z] + \frac{d}{d z} [- 10]$

Step 2.4.2

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

$- 12 z^{2} + 18 z + 4 \cdot 1 + \frac{d}{d z} [- 10]$

Step 2.4.3

Multiply $4$ by $1$ .

$- 12 z^{2} + 18 z + 4 + \frac{d}{d z} [- 10]$

Step 2.5

Differentiate using the Constant Rule.

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Step 2.5.1

Since $- 10$ is constant with respect to $z$ , the derivative of $- 10$ with respect to $z$ is $0$ .

$- 12 z^{2} + 18 z + 4 + 0$

Step 2.5.2

Add $- 12 z^{2} + 18 z + 4$ and $0$ .

$f'' (z) = - 12 z^{2} + 18 z + 4$

Step 3

The second derivative of $h (z)$ with respect to $z$ is $- 12 z^{2} + 18 z + 4$ .

$- 12 z^{2} + 18 z + 4$